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Constrained characteristic functions, multivariable interpolation, and invariant subspaces

Abstract

In this paper, we present a functional model theorem for completely non-coisometric n-tuples of operators in the noncommutative variety \(\mathcal{V}_{f,\varphi,\mathcal{I}}(\mathcal{H})\) in terms of constrained characteristic functions. As an application, we prove that the constrained characteristic function is a complete unitary invariant for this class of elements, which can be viewed as the noncommutative analogue of the classical Sz.-Nagy–Foiaş functional model for completely nonunitary contractions. On the other hand, we provide a Sarason-type commutant lifting theorem. Applying this result, we solve the Nevanlinna–Pick-type interpolation problem in our setting. Moreover, we also obtain a Beurling-type characterization of the joint invariant subspaces under the operators \(B_{1},\ldots,B_{n}\), where the n-tuple \((B_{1},\ldots,B_{n})\) is the universal model associated with the abstract noncommutative variety \(\mathcal{V}_{f,\varphi,\mathcal{I}}\).

Introduction

In the last fifty years, the study of the closed operator unit ball

$$\bigl[B(\mathcal{H})\bigr]^{-}_{1}:=\bigl\{ T\in B(\mathcal{H}): \bigl\Vert TT^{*} \bigr\Vert ^{\frac{1}{2}}\le1\bigr\} $$

has generated the celebrated Sz.-Nagy–Foiaş theory of contractions on Hilbert spaces. This research has evolved into a well-developed theory, which plays an important role in modern functional analysis. In 1963, Sz.-Nagy and Foiaş obtained an effective \(H^{\infty}\)-functional calculus for completely nonunitary contractions on Hilbert spaces based on the existence of a unitary dilation of a contraction T (see [33]). An important application of this functional calculus to the theory of contraction semigroups has also been given in Foiaş [5]. Moreover, the characteristic function of a contraction T appears as the operator-valued analytic function corresponding to a certain orthogonal projection in the space of the minimal unitary dilation of T. This yields a functional model for T, which is a useful tool for analyzing the structure of contractions.

In the multivariable case, the study of the closed operator unit n-ball

$$\bigl[B(\mathcal{H})^{n}\bigr]^{-}_{1}:=\bigl\{ (T_{1},\ldots,T_{n})\in B(\mathcal{H})^{n}: \bigl\Vert T_{1}T_{1}^{*}+\cdots+T_{n}T_{n}^{*} \bigr\Vert ^{\frac{1}{2}}\le1\bigr\} $$

has generated a noncommutative analogue of Sz.-Nagy–Foiaş theory (see [24, 68], and more recently [1, 11, 34]). In particular, Popescu developed a theory of holomorphic functions in several noncommuting variables and provided a framework for the study of arbitrary n-tuples of operators. A free analytic functional calculus was introduced and studied in connection with Hausdorff derivations, noncommutative Cauchy and Poisson transforms, and von Neumann inequalities (see [15, 16, 18, 2023, 26, 29, 30]). Moreover, we remark the work of Helton, McCullough, and Vinnikov on symmetric noncommutative polynomials (see [9, 10]). We should also remark that, in recent years, many results concerning the theory of row contractions were extended by Muhly and Solel ([1214]) to representations of tensor algebras over \(C^{*}\)-correspondences and Hardy algebras.

In [28], Popescu developed an operator model theory for pure n-tuples of operators in noncommutative domains \(\mathbb{D}_{f,\varphi }(\mathcal{H})\subset B(\mathcal{H})^{n}\) generated by positive regular free holomorphic functions f and certain classes of n-tuples \(\varphi=(\varphi _{1},\ldots,\varphi_{n})\) of formal power series in noncommutative indeterminates \(Z_{1},\ldots,Z_{n}\). An important role in his study was played by noncommutative Poisson transforms. Using these transforms, he proved that each abstract noncommutative domain \(\mathbb{D}_{f,\varphi}\) has a universal model \((M_{Z_{1}},\ldots,M_{Z_{n}})\). Unlike the case of the ball \([B(\mathcal{H})^{n}]_{1}^{-}\), the operators \(M_{Z_{1}},\ldots,M_{Z_{n}}\) are not isometries and do not have orthogonal ranges in general, which leads to considerable technical difficulties in developing an operator model theory. Moreover, notice that the study of \(\mathbb{D}_{f,\varphi }(\mathcal{H})\) is closely related to the study of the operators \(M_{Z_{1}},\ldots,M_{Z_{n}}\), their joint invariant subspaces, and the representations of the algebras they generate: the noncommutative domain algebra \(\mathcal{A}(\mathbb{D}_{f,\varphi})\), the noncommutative Hardy algebra \(H^{\infty}(\mathbb{D}_{f,\varphi})\), and the \(C^{*}\)-algebra \(C^{*}(M_{Z_{1}},\ldots,M_{Z_{n}})\). Indeed, this noncommutative domain \(\mathbb{D}_{f,\varphi}(\mathcal{H})\) has been studied in several particular cases. According to [22, 24] and [33], if \(f=Z\) and \(\varphi=Z\), then the corresponding domain \(\mathbb{D}_{f,\varphi}(\mathcal{H})\) coincides with the closed operator unit ball \([B(\mathcal{H})]_{1}^{-}\), the study of which has generated Sz.-Nagy–Foiaş theory of contractions. If \(f=Z_{1}+\cdots+Z_{n}\) and \(\varphi=(Z_{1},\ldots,Z_{n})\), then the corresponding domain \(\mathbb{D}_{f,\varphi}(\mathcal{H})\) coincides with the closed operator unit n-ball \([B(\mathcal {H})^{n}]_{1}^{-}\), the study of which has generated a free analogue of Sz.-Nagy–Foiaş theory. In particular, if \(\varphi=(Z_{1},\ldots ,Z_{n})\), then the corresponding domain \(\mathbb{D}_{f,\varphi}(\mathcal {H})\) coincides with the noncommutative Reinhardt domain \(\mathcal {D}_{f}(\mathcal{H})\), which was first studied by Popescu [24].

In this paper, we continue the research line of Popescu to develop an operator model theory for completely non-coisometric n-tuples of operators in noncommutative varieties \(\mathcal{V}_{f,\varphi,\mathcal {I}}(\mathcal{H})\). To present our results, we need some notation. Let \(\mathbf{S}[Z_{1},\ldots,Z_{n}]\) be the algebra of all formal power series in noncommutative indeterminates \(Z_{1},\ldots,Z_{n}\) and complex coefficients. We denote by \(\mathbb{F}_{n}^{+}\) the unital free semigroup on n generators \(g_{1},\ldots ,g_{n}\) and the identity \(g_{0}\). The length of \(\alpha\in\mathbb{F}_{n}^{+}\) is defined by \(|\alpha|:=0\) if \(\alpha=g_{0}\) and \(|\alpha|:=k\) if \(\alpha =g_{i_{1}}\cdots g_{i_{k}}\), where \(i_{1},\ldots,i_{k}\in\{1,\ldots,n\}\). We set \(Z_{\alpha}:=Z_{i_{1}}\cdots Z_{i_{k}}\) and \(Z_{g_{0}}:= I\). If \(f\in\mathbf{S}[Z_{1},\ldots,Z_{n}]\) has the representation \(f:=\sum_{\alpha\in\mathbb{F}_{n}^{+}}a_{\alpha}Z_{\alpha}\) and the coefficients \(a_{\alpha}\in\mathbb{C}\) satisfy the conditions

$$r(f)^{-1}:=\limsup_{k\to\infty}\biggl(\sum _{ \vert \alpha \vert =k} \vert a_{\alpha} \vert ^{2} \biggr)^{\frac {1}{2k}}< \infty, $$

\(a_{\alpha}\ge0\) for any \(\alpha\in\mathbb{F}_{n}^{+}\), \(a_{g_{0}}=0\), and \(a_{g_{i}}>0\), \(i=1,\ldots,n\), we say that f is a positive regular free holomorphic function. The number \(r(f)\) is called the radius of convergence of f.

Denote by \(\mathcal{M}_{f}\) the set of all n-tuples \(\varphi=(\varphi _{1},\ldots,\varphi_{n})\) of formal power series \(\varphi_{i}\in\mathbf {S}[Z_{1},\ldots, Z_{n}]\) with the model property (see Sect. 2). \(\mathcal{H}\) is a Hilbert space and \(B(\mathcal{H})\) is the algebra of all bounded linear operators on \(\mathcal{H}\). If \(X = (X_{1},\ldots, X_{n}) \in B(\mathcal{H})^{n}\), we denote \(X_{\alpha}:=X_{i_{1}}\cdots X_{i_{k}}\) if \(\alpha=g_{i_{1}}\cdots g_{i_{k}}\in\mathbb{F}_{n}^{+}\), and \(X_{g_{0}} := I_{\mathcal{H}}\). We introduce the noncommutative domain \(\mathbb {D}_{f,\varphi}(\mathcal{H})\) associated with \(f,\varphi\in\mathcal{M}_{f}\) and a Hilbert space \(\mathcal{H}\) and defined by

$$\mathbb{D}_{f,\varphi}(\mathcal{H}):=\biggl\{ X\in B(\mathcal{H})^{n}: \psi \bigl(\varphi(X)\bigr)=X \mbox{ and } \sum_{|\alpha|\ge1}a_{\alpha}\bigl[\varphi(X)\bigr]_{\alpha}\bigl[\varphi(X)\bigr]_{\alpha}^{*} \le I_{\mathcal{H}}\biggr\} , $$

where \(\psi:=(\psi_{1},\ldots,\psi_{n})\) is the inverse of φ with respect to composition of formal power series, and the evaluations are well defined (see Sect. 2). We refer to \(\mathbb{D}_{f,\varphi }:=\{\mathbb{D}_{f,\varphi}(\mathcal{H}): \mathcal{H}\text{ is a Hilbert space}\}\) as the abstract noncommutative domain, and to \(\mathbb {D}_{f,\varphi}(\mathcal{H})\) as its representation on the Hilbert space \(\mathcal{H}\). We associate with each \(\mathbb{D}_{f,\varphi}\) a Hilbert space \(\mathbb {H}_{f}^{2}(\varphi)\) of formal power series in \(\mathbf{S}[Z_{1},\ldots, Z_{n}]\) with the property that the indeterminates \(Z_{1},\ldots, Z_{n}\) are in the Hilbert space \(\mathbb{H}_{f}^{2}(\varphi)\) and each left multiplication operator \(M_{Z_{i}}:\mathbb{H}_{f}^{2}(\varphi )\to\mathbb{H}_{f}^{2}(\varphi)\) defined by

$$M_{Z_{i}}\zeta:=Z_{i}\zeta,\quad\zeta\in \mathbb{H}_{f}^{2}(\varphi), $$

is a bounded multiplier of \(\mathbb{H}_{f}^{2}(\varphi)\). Similarly, each right multiplication operator \(R_{Z_{i}}:\mathbb{H}_{f}^{2}(\varphi)\to\mathbb {H}_{f}^{2}(\varphi)\) defined by

$$R_{Z_{i}}\zeta:=\zeta Z_{i},\quad\zeta\in \mathbb{H}_{f}^{2}(\varphi), $$

is also a bounded multiplier of \(\mathbb{H}_{f}^{2}(\varphi)\).

Let \(\mathcal{I}\neq H^{\infty}(\mathbb{D}_{f,\varphi})\) be a WOT-closed two-sided ideal of the noncommutative Hardy algebra \(H^{\infty}(\mathbb{D}_{f,\varphi})\), where \(H^{\infty}(\mathbb {D}_{f,\varphi})\) is the WOT-closure of all noncommutative polynomials in \(M_{Z_{1}},\ldots,M_{Z_{n}}\) and the identity. Now we define the noncommutative variety

$$\mathcal{V}_{f,\varphi,\mathcal{I}}(\mathcal{H}):=\bigl\{ (X_{1}, \ldots,X_{n})\in \mathbb{D}_{f,\varphi}(\mathcal{H}): \omega(X_{1},\ldots,X_{n})=0 \mbox{ for any } \omega\in \mathcal{I}\bigr\} . $$

Denote by \(H^{\infty}(\mathcal{V}_{f,\varphi,\mathcal{I}})\) the WOT-closed algebra generated by the constrained weighted shifts \(B_{i}:=P_{\mathcal{N}_{f,\varphi,\mathcal{I}}}M_{Z_{i}}|_{\mathcal {N}_{f,\varphi,\mathcal{I}}}\) for \(i=1,\ldots,n\) and the identity, where

$$\mathcal{N}_{f,\varphi,\mathcal{I}}:=\mathbb{H}_{f}^{2}( \varphi)\ominus \mathcal{M}_{f,\varphi,\mathcal{I}}\quad\mbox{and}\quad\mathcal {M}_{f,\varphi,\mathcal{I}} :=\overline{\mathcal{I}\mathbb{H}_{f}^{2}( \varphi)}. $$

Similarly, denote by \(R^{\infty}(\mathcal{V}_{f,\varphi,\mathcal{I}})\) the WOT-closed algebra generated by the constrained weighted shifts \(C_{i}:=P_{\mathcal{N}_{f,\varphi,\mathcal{I}}}R_{Z_{i}}|_{\mathcal {N}_{f,\varphi,\mathcal{I}}}\) for \(i=1,\ldots,n\) and the identity.

In Sect. 2, we collect some notation and preliminaries which are needed in the sequel. In Sect. 3, we obtain a factorization result for the constrained characteristic function, namely

$$I_{\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{D}_{C_{f,\varphi ,T}}}-\varTheta_{f,\varphi,T}^{(\mathcal{I})}\bigl( \varTheta_{f,\varphi ,T}^{(\mathcal{I})}\bigr)^{*}= K_{f,\varphi,T}^{(\mathcal{I})} \bigl(K_{f,\varphi,T}^{(\mathcal{I})}\bigr)^{*}, $$

where \(\varTheta_{f,\varphi,T}^{(\mathcal{I})}\) is the constrained characteristic function and \(K_{f,\varphi,T}^{(\mathcal{I})}\) is the corresponding constrained Poisson kernel. Moreover, we present a functional model theorem for completely non-coisometric n-tuples of operators in the noncommutative variety \(\mathcal{V}_{f,\varphi,\mathcal{I}}(\mathcal{H})\) in terms of constrained characteristic functions. Applying this result, we prove that the constrained characteristic function is a complete unitary invariant for this class of elements. Indeed, this result can be viewed as the noncommutative analogue of the classical Sz.-Nagy–Foiaş functional model for completely nonunitary contractions.

In Sect. 4, we prove a Sarason-type commutant lifting theorem. As an application, we obtain the Nevanlinna–Pick-type interpolation result in our setting. We show that if \(\lambda_{1},\ldots,\lambda_{k}\) are k distinct points in the strict noncommutative variety \(\mathcal{V}_{f,\varphi,\mathcal {I}}^{<}(\mathbb{C})\) and \(A_{1},\ldots,A_{k}\in B(\mathcal{K})\), then there exists \(\varPhi(C_{1},\ldots,C_{n})\in R^{\infty}(\mathcal{V}_{f,\varphi ,\mathcal{I}})\mathrel{\overline{\otimes}}B(\mathcal{K})\) such that

$$\bigl\Vert \varPhi(C_{1},\ldots,C_{n}) \bigr\Vert \le1\quad \mbox{and}\quad \varPhi(\lambda _{j})=A_{j}, \quad j=1,\ldots,k, $$

if and only if the operator matrix

$$\bigl[K_{f,\varphi}(\lambda_{i},\lambda_{j}) \bigl(I_{\mathcal{K}}-A_{i}A_{j}^{*}\bigr) \bigr]_{k\times k} $$

is positive semidefinite, where

$$K_{f,\varphi}(\lambda_{i},\lambda_{j}):= \frac{\sqrt{1-\sum_{ \vert \alpha \vert \ge 1}a_{\alpha} \vert \varphi_{\alpha}(\lambda_{i}) \vert ^{2}}\sqrt{1-\sum_{ \vert \alpha \vert \ge 1}a_{\alpha} \vert \varphi_{\alpha}(\lambda_{j}) \vert ^{2}}}{ 1-\sum_{ \vert \alpha \vert \ge1}a_{\alpha}[\varphi(\lambda_{i})]_{\alpha}[\overline {\varphi(\lambda_{j})}]_{\alpha}}. $$

Moreover, we provide a Beurling-type characterization of the joint invariant subspaces under the constrained weighted shifts \(B_{1},\ldots ,B_{n}\). More precisely, a subspace \(\mathcal{M}\subseteq\mathcal{N}_{f,\varphi,\mathcal {I}}\otimes\mathcal{K}\) is invariant under \(B_{i}\otimes I_{\mathcal{K}}\), \(i=1,\ldots,n\), if and only if there are a Hilbert space \(\mathcal{G}\) and an inner multi-analytic operator

$$\varPhi:\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{G}\to\mathcal {N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{K} $$

with respect to the constrained weighted shifts \(B_{1},\ldots,B_{n}\) such that

$$\mathcal{M}=\varPhi[\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{G}]. $$

Preliminaries

In this section we collect some notation and preliminaries which are needed in the sequel. For more information, we refer to [24, 27] and [28].

Weighted Fock space

Let \(f:=\sum_{\alpha\in\mathbb{F}_{n}^{+}}a_{\alpha}Z_{\alpha}\), \(a_{\alpha}\in \mathbb{C}\), be a positive regular free holomorphic function. Define the noncommutative domain

$$\mathcal{D}_{f}(\mathcal{H}):=\biggl\{ (X_{1}, \ldots,X_{n})\in B(\mathcal{H})^{n}:\sum _{|\alpha|\ge1}a_{\alpha}X_{\alpha}X_{\alpha}^{*}\le I_{\mathcal{H}}\biggr\} , $$

where the convergence of the series is in the weak operator topology. Define the strict noncommutative domain

$$\mathcal{D}_{f,< }(\mathcal{H}):=\biggl\{ (X_{1}, \ldots,X_{n})\in B(\mathcal {H})^{n}: \biggl\Vert \sum _{ \vert \alpha \vert \ge1}a_{\alpha}X_{\alpha}X_{\alpha}^{*} \biggr\Vert < 1\biggr\} , $$

where the convergence is in the weak operator topology. Now, we define

$$ b_{g_{0}}=1\quad \mbox{and}\quad b_{\alpha}= \sum_{j=1}^{ \vert \alpha \vert }\sum _{\substack{\gamma_{1} \cdots\gamma_{j}=\alpha\\ \vert \gamma_{1} \vert \ge1,\ldots , \vert \gamma_{j} \vert \ge1}}a_{\gamma_{1}} \cdots a_{\gamma_{j}}\quad \mbox{if } \vert \alpha \vert \ge1. $$
(2.1)

We introduce an inner product on the algebra of noncommutative polynomials \(\mathbb{C}[Z_{1},\ldots,Z_{n}]\) by setting

$$\langle Z_{\alpha},Z_{\beta}\rangle_{f}:= \frac{1}{b_{\alpha}}\delta_{\alpha\beta },\quad\alpha,\beta\in \mathbb{F}_{n}^{+}. $$

Let \(\mathcal{F}_{f}^{2}\) be the completion of \(\mathbb{C}[Z_{1},\ldots,Z_{n}]\) in this inner product. Notice that the elements of \(\mathcal{F}_{f}^{2}\) are formal power series \(\zeta\in\mathbf{S}[Z_{1},\ldots,Z_{n}]\) of the form \(\zeta=\sum_{\alpha \in\mathbb{F}_{n}^{+}}c_{\alpha}Z_{\alpha}\), where

$$\Vert \zeta \Vert _{f}^{2}:=\sum _{\alpha\in\mathbb{F}_{n}^{+}} \vert c_{\alpha} \vert ^{2} \frac {1}{b_{\alpha}}< \infty. $$

Indeed, \(\mathcal{F}_{f}^{2}\) is a weighted Fock space on n generators. For each \(i=1,\ldots,n\), we define the left multiplication operator \(V_{i}:\mathcal{F}_{f}^{2}\to\mathcal{F}_{f}^{2}\) by setting \(V_{i}\zeta:=Z_{i}\zeta\). Notice that \((V_{1},\ldots,V_{n})\) is in the noncommutative domain \(\mathcal {D}_{f}(\mathcal{F}_{f}^{2})\), and

$$ I_{\mathcal{F}_{f}^{2}}-\sum_{|\alpha|\ge1}a_{\alpha}V_{\alpha}V_{\alpha}^{*}=P_{\mathbb{C}}, $$
(2.2)

where \(P_{\mathbb{C}}\) is the orthogonal projection from \(\mathcal {F}_{f}^{2}\) onto \(\mathbb{C}\).

Let \(\mathcal{F}_{f}^{\infty}\) be the set of all \(\zeta\in\mathcal{F}_{f}^{2}\) with the property that

$$\Vert \zeta \Vert _{\infty}:=\sup\bigl\{ \Vert \zeta p \Vert _{f}:p\in\mathbb{C}[Z_{1},\ldots,Z_{n}], \Vert p \Vert _{f}\le1\bigr\} < \infty. $$

Notice that \(\mathcal{F}_{f}^{\infty}\) is a Banach algebra with respect to the norm \(\|\cdot\|_{\infty}\). Let \(\zeta=\sum_{\beta\in\mathbb {F}_{n}^{+}}c_{\beta}Z_{\beta}\) be a formal power series with the property that \(\sum_{\beta\in\mathbb {F}_{n}^{+}}|c_{\beta}|^{2} \frac{1}{b_{\beta}}<\infty\), where the coefficients \(b_{\beta}\), \(\beta\in\mathbb{F}_{n}^{+}\), are given by relation (2.1). One can see that \(\sum_{\beta\in \mathbb{F}_{n}^{+}}c_{\beta}V_{\beta}(p)\in\mathcal{F}_{f}^{2}\) for any \(p\in \mathbb{C}[Z_{1},\ldots,Z_{n}]\). Moreover, \(\zeta\in\mathcal{F}_{f}^{\infty}\) if and only if

$$\sup_{p\in\mathbb{C}[Z_{1},\ldots,Z_{n}], \Vert p \Vert _{f}\le1} \biggl\Vert \sum_{\beta\in\mathbb {F}_{n}^{+}}c_{\beta}V_{\beta}(p) \biggr\Vert _{f}< \infty. $$

In this case, there is a unique bounded operator acting on \(\mathcal {F}_{f}^{2}\), which we denote by \(\zeta(V_{1},\ldots,V_{n})\), such that

$$\zeta(V_{1},\ldots,V_{n})p=\sum _{\beta\in\mathbb{F}_{n}^{+}}c_{\beta}V_{\beta}(p)\quad \mbox{for any } p\in\mathbb{C}[Z_{1},\ldots,Z_{n}]. $$

We call the series \(\sum_{\beta\in\mathbb{F}_{n}^{+}}c_{\beta}V_{\beta}\) the Fourier representation of \(\zeta(V_{1},\ldots,V_{n})\). The set of all operators \(\varphi(V_{1},\ldots,V_{n})\in B(\mathcal {F}_{f}^{2})\) satisfying the above-mentioned properties is denoted by \(F^{\infty}(\mathcal{D}_{f})\).

We consider the full Fock space of \(H_{n}\) defined by

$$F^{2}(H_{n}):=\mathbb{C}1\oplus\bigoplus _{m\geq1}H_{n}^{\otimes m}, $$

where \(H_{n}^{\otimes m}\) is the Hilbert tensor product of m copies of \(H_{n}\). We denote \(e_{\alpha}:=e_{i_{1}}\otimes\cdots\otimes e_{i_{k}}\) if \(\alpha=g_{i_{1}}\cdots g_{i_{k}}\), where \(i_{1},\ldots,i_{k}\in\{1,\ldots ,n\}\), and \(e_{g_{0}}:=1\). Consider \(\varOmega:F^{2}(H_{n})\to\mathcal{F}_{f}^{2}\) to be the unitary operator defined by \(\varOmega(e_{\alpha}):=\sqrt{b_{\alpha}}Z_{\alpha}\), \(\alpha\in\mathbb{F}_{n}^{+}\), where the coefficients \(b_{\alpha}\) are given by relation (2.1). We remark that \(\varOmega ^{-1}V_{i}\varOmega=W_{i}\), \(i=1,\ldots,n\), where \((W_{1},\ldots,W_{n})\) is the n-tuple of weighted shifts on \(F^{2}(H_{n})\), which was introduced in [24]. Using the results from [24], we know that \(F^{\infty}(\mathcal{D}_{f})\) is the WOT-closure (resp. SOT-closure, \(w^{*}\)-closure) of all polynomials in \(V_{1},\ldots ,V_{n}\) and the identity. The noncommutative domain algebra \(\mathcal {A}(\mathcal{D}_{f})\) is the norm-closure of all polynomials in \(V_{1},\ldots,V_{n}\) and the identity.

Noncommutative domain

We say that an n-tuple \(p=(p_{1},\ldots,p_{n})\) of polynomials is invertible with respect to composition if there exists an n-tuple \(q=(q_{1},\ldots,q_{n})\) of polynomials such that \(p\circ q=q\circ p=\mathit{id}\). In this case, we say that p has property (\(\mathcal{A}\)). In what follows, we provide an example. If

$$\begin{aligned}& p_{1}= {a_{1}Z_{1}+a_{2}Z_{2}+a_{3}Z_{3}Z_{2},} \\& p_{2}= {b_{2}Z_{2}+b_{3}Z_{3}^{2} \quad(a_{1}b_{2}c_{3}\neq0),} \\& p_{3}= {c_{3}Z_{3},} \end{aligned}$$

then \(p=(p_{1},p_{2},p_{3})\) is invertible with respect to composition, i.e., there exists \(q=(q_{1},q_{2},q_{3})\) such that \(p\circ q=q\circ p=\mathit{id}\), where

$$\begin{aligned}& q_{1}= {\frac{1}{a_{1}}Z_{1}- \frac{a_{2}}{a_{1} b_{2}}Z_{2}-\frac{a_{3}}{a_{1}b_{2} c_{3}}Z_{3}Z_{2}+ \frac{a_{2} b_{3}}{a_{1} b_{2}c_{3}^{2}}Z_{3}^{2}+\frac{a_{3}b_{3}}{a_{1}b_{2}c_{3}^{3}} Z_{3}^{3},} \\& q_{2} = {\frac{1}{b_{2}}Z_{2}- \frac{b_{3}}{b_{2}c_{3}^{2}}Z_{3}^{2},} \\& q_{3} = {\frac{1}{c_{3}} Z_{3}.} \end{aligned}$$

This shows that p has property (\(\mathcal{A}\)).

Let \(f:=\sum_{\alpha\in\mathbb{F}_{n}^{+}}a_{\alpha}Z_{\alpha}\) be a positive regular free holomorphic function, and let \(p=(p_{1},\ldots ,p_{n})\) be an n-tuple of noncommutative polynomials with property (\(\mathcal{A}\)). We introduce an inner product by setting

$$\langle p_{\alpha},p_{\beta}\rangle_{f,p}:= \frac{1}{b_{\alpha}}\delta_{\alpha \beta},\quad\alpha,\beta\in \mathbb{F}_{n}^{+}. $$

Let \(\mathbb{H}_{f}^{2}(p)\) be the completion of the linear space \(\bigvee\{ p_{\alpha}\}_{\alpha\in\mathbb{F}_{n}^{+}}\) with respect to this inner product.

Consider an n-tuple of formal power series \(\varphi=(\varphi _{1},\ldots,\varphi_{n})\) in indeterminates \(Z_{1},\ldots,Z_{n}\) with the property that the Jacobian

$${\operatorname{det}J_{\varphi}(0):=\operatorname{det}[\lambda _{ij}]_{i,j=1}^{n}\neq0,} $$

where

$${\varphi_{i}(Z_{1},\ldots,Z_{n})=a_{0}^{(i)}I+ \sum_{p=1}^{n}a_{p}^{(i)}Z_{p}+ \sum_{|\alpha|\ge2}a_{\alpha}^{(i)}Z_{\alpha}, \lambda_{ij}=a_{j}^{(i)},} $$

and \(i,j=1,\ldots,n\). Due to Theorem 1.2 from [25], the set \(\{\varphi_{\alpha}\}_{\alpha \in\mathbb{F}_{n}^{+}}\) (where \(\varphi_{0}:=I\)) is linearly independent in \(\mathbf{S}[Z_{1},\ldots,Z_{n}]\). We introduce an inner product on the linear span of \(\{\varphi_{\alpha}\} _{\alpha\in\mathbb{F}_{n}^{+}}\) by setting

$$\langle\varphi_{\alpha},\varphi_{\beta}\rangle_{f,\varphi}:= \frac {1}{b_{\alpha}}\delta_{\alpha\beta},\quad\alpha,\beta\in \mathbb{F}_{n}^{+}, $$

where the coefficients \(b_{\alpha}\), \(\alpha\in\mathbb{F}_{n}^{+}\), are given by relation (2.1). Let \(\mathbb{H}_{f}^{2}(\varphi)\) be the completion of the linear space \(\bigvee\{\varphi_{\alpha}\}_{\alpha\in\mathbb{F}_{n}^{+}}\) with respect to this inner product. Assume now that \(\varphi(0)=0\). Theorem 1.3 from [25] shows that φ is not a right zero divisor with respect to composition, i.e., there is no nonzero power series χ in \(\mathbf{S}[Z_{1},\ldots,Z_{n}]\) such that \(\chi\circ\varphi=0\). Consequently, the elements of \(\mathbb{H}_{f}^{2}(\varphi)\) can be seen as a formal power series in \(\mathbf{S}[Z_{1},\ldots,Z_{n}]\) of the form \(\sum_{\alpha\in\mathbb{F}_{n}^{+}}c_{\alpha}\varphi_{\alpha}\), where \(\sum_{\alpha\in\mathbb{F}_{n}^{+}}\frac{1}{b_{\alpha}}|c_{\alpha}|^{2}<\infty\).

To introduce the class of n-tuples of formal power series with property \((\mathcal{S})\), we need some preliminaries. Let \(\chi=\sum_{k=0}^{\infty}\sum_{|\alpha|=k}c_{\alpha}Z_{\alpha}\) be a formal power series in indeterminates \(Z_{1},\ldots,Z_{n}\). We denote by \(\mathcal {C}_{\chi}(\mathcal{H})\) (resp. \(\mathcal{C}_{\chi}^{\mathrm{SOT}} (\mathcal{H})\)) the set of all \(Y:=(Y_{1},\ldots,Y_{n})\in B(\mathcal{H})^{n}\) such that the series \(\chi (Y_{1},\ldots,Y_{n}):= \sum_{k=0}^{\infty}\sum_{|\alpha|=k}c_{\alpha}Y_{\alpha}\) is norm (resp. SOT) convergent. These sets are called sets of norm (resp. SOT) convergence for the power series χ. We also introduce the set \(\mathcal{C}_{\chi}^{\mathrm{rad}} (\mathcal{H})\) of all \(Y:=(Y_{1},\ldots,Y_{n})\in B(\mathcal{H})^{n}\) such that there exists \(\delta \in(0,1)\) with the property that \(rY\in\mathcal{C}_{\chi}(\mathcal{H})\) for any \(r\in (\delta,1)\) and

$$\widehat{\chi}(Y_{1},\ldots,Y_{n}):=\text{SOT-} \lim_{r\to1}\sum_{k=0}^{\infty}\sum_{|\alpha|=k}c_{\alpha}r^{|\alpha|}Y_{\alpha}$$

exists.

Definition 2.1

(see [28])

Let \(\varphi=(\varphi_{1},\ldots,\varphi_{n})\) be an n-tuple of formal power series in \(Z_{1},\ldots,Z_{n}\) such that \(\varphi(0)=0\). We say that φ has property (\(\mathcal{S}\)) if the following conditions hold:

(\(\mathcal{S}_{1}\)):

The radius of convergence of φ, i.e., \(r(\varphi):=\min_{i=1,\ldots,n}r(\varphi_{i})\), is strictly positive and det \(J_{\varphi}(0)\neq0\).

(\(\mathcal{S}_{2}\)):

The indeterminates \(Z_{1},\ldots,Z_{n}\) are in the Hilbert space \(\mathbb{H}_{f}^{2}(\varphi)\) and each multiplication operator \(M_{Z_{i}}:\mathbb{H}_{f}^{2}(\varphi)\to\mathbb{H}_{f}^{2}(\varphi)\) defined by

$$M_{Z_{i}}\zeta:=Z_{i}\zeta,\quad\zeta\in \mathbb{H}_{f}^{2}(\varphi), $$

is a bounded multiplier of \(\mathbb{H}_{f}^{2}(\varphi)\).

(\(\mathcal{S}_{3}\)):

The multiplication operators \(M_{\varphi _{j}}:\mathbb{H}_{f}^{2}(\varphi)\to\mathbb{H}_{f}^{2}(\varphi)\), \(M_{\varphi _{j}}\chi=\varphi_{j}\chi\), satisfy the equations

$$M_{\varphi_{j}}=\varphi_{j}(M_{Z_{1}}, \ldots,M_{Z_{n}}),\quad j=1,\ldots,n, $$

where \((M_{Z_{1}},\ldots,M_{Z_{n}})\) is either in the convergence set \(\mathcal{C}_{\varphi}^{\mathrm{SOT}}(\mathbb{H}_{f}^{2}(\varphi))\) or \(\mathcal {C}_{\varphi}^{\mathrm{rad}}(\mathbb{H}_{f}^{2}(\varphi))\).

Let \(U:\mathbb{H}_{f}^{2}(\varphi)\to\mathcal{F}_{f}^{2}\) be the unitary operator defined by \(U(\varphi_{\alpha}):=Z_{\alpha}\), \(\alpha \in\mathbb{F}_{n}^{+}\). According to the proof of Lemma 1.2 from [28], we have

$$ M_{\varphi_{i}}=U^{-1}V_{i}U,\quad i=1,\ldots,n. $$
(2.3)

Throughout this paper, unless otherwise specified, we assume that \(\varphi=(\varphi_{1},\ldots, \varphi_{n})\) is either an n-tuple of noncommutative polynomials with property (\(\mathcal{A}\)) or an n-tuple of formal power series with \(\varphi(0)=0\) and property (\(\mathcal{S}\)). In this case, we say that φ has the model property.

Definition 2.2

(see [25, 28])

Let \(\varphi=(\varphi_{1},\ldots,\varphi_{n})\) be an n-tuple of formal power series with model property, and let \(\psi=(\psi_{1},\ldots,\psi_{n})\) be the n-tuple of power series which is the inverse of \(\varphi =(\varphi_{1},\ldots,\varphi_{n})\) with respect to composition. Assume that \(\psi_{i}\) has the representation

$$\psi_{i}=\sum_{k=0}^{\infty}\sum_{\alpha\in\mathbb{F}_{n}^{+},|\alpha |=k}c_{\alpha}^{(i)}Z_{\alpha}\quad\mbox{for } i=1,\ldots,n, $$

where the sequence \(\{c_{\alpha}^{(i)}\}_{\alpha\in\mathbb{F}_{n}^{+}}\) is uniquely determined by the condition \(\psi\circ\varphi=\mathit{id}\). We say that an n-tuple of operators \(X=(X_{1},\ldots,X_{n})\in B(\mathcal {H})^{n}\) satisfies the equation \(\psi(\varphi(X))=X\) in either one of the following two cases:

  1. (a)

    \(X\in\mathcal{C}_{\varphi}^{\mathrm{SOT}}(\mathcal{H})\) and either \(X_{i}=\sum_{k=0}^{\infty}\sum_{\alpha\in\mathbb{F}_{n}^{+},|\alpha|=k}c_{\alpha}^{(i)}[\varphi(X)]_{\alpha}\), \(i=1,\ldots,n\), where the convergence of the series is in the strong operator topology, or \(\varphi(X)\in\mathcal{C}_{\psi}^{\mathrm{rad}}(\mathcal {H})\) and

    $$X_{i}=\text{SOT-}\lim_{r\to1}\sum _{k=0}^{\infty}\sum_{\alpha\in\mathbb {F}_{n}^{+},|\alpha|=k}c_{\alpha}^{(i)}r^{|\alpha|} \bigl[\varphi(X)\bigr]_{\alpha},\quad i=1,\ldots,n; $$
  2. (b)

    \(X\in\mathcal{C}_{\varphi}^{\mathrm{rad}}(\mathcal{H})\) and either \(X_{i}=\sum_{k=0}^{\infty}\sum_{\alpha\in\mathbb{F}_{n}^{+},|\alpha|=k}c_{\alpha}^{(i)}[\widehat{\varphi}(X)]_{\alpha}\), \(i=1,\ldots,n\), where the convergence of the series is in the strong operator topology, or \(\widehat{\varphi}(X)\in\mathcal{C}_{\psi}^{\mathrm{rad}}(\mathcal{H})\) and

    $$X_{i}=\text{SOT-}\lim_{r\to1}\sum _{k=0}^{\infty}\sum_{\alpha\in\mathbb {F}_{n}^{+},|\alpha|=k}c_{\alpha}^{(i)}r^{|\alpha|} \bigl[\widehat{\varphi }(X)\bigr]_{\alpha},\quad i=1,\ldots,n. $$

Definition 2.3

(see [28])

Let \(f:=\sum_{\alpha\in\mathbb{F}_{n}^{+}}a_{\alpha}Z_{\alpha}\) be a positive regular free holomorphic function, and let \(\varphi=(\varphi_{1},\ldots ,\varphi_{n})\) be an n-tuple of formal power series with model property. The noncommutative domain \(\mathbb{D}_{f,\varphi}(\mathcal{H})\) is the set of all n-tuples of bounded linear operators \(X=(X_{1},\ldots,X_{n})\in B(\mathcal{H})^{n}\) such that \(\psi(\varphi(X))=X\) and

$$\sum_{|\alpha|\ge1}a_{\alpha}\bigl[\varphi(X) \bigr]_{\alpha}\bigl[\varphi(X)\bigr]_{\alpha}^{*}\le I_{\mathcal{H}}, $$

where the convergence is in the weak operator topology. Define the strict noncommutative domain

$$\mathbb{D}_{f,\varphi}^{< }(\mathcal{H}):=\biggl\{ X\in B(\mathcal{H})^{n}: \psi \bigl(\varphi(X)\bigr)=X \mbox{ and } \biggl\Vert \sum _{ \vert \alpha \vert \ge1}a_{\alpha}\bigl[\varphi (X) \bigr]_{\alpha}\bigl[\varphi(X)\bigr]_{\alpha}^{*} \biggr\Vert < 1\biggr\} , $$

where the convergence is in the weak operator topology.

We define the noncommutative Hardy algebra \(H^{\infty}(\mathbb {D}_{f,\varphi})\) to be the WOT-closure of all noncommutative polynomials in \(M_{Z_{1}},\ldots,M_{Z_{n}}\) and the identity. Similarly, we can also define the noncommutative Hardy algebra \(R^{\infty}(\mathbb{D}_{f,\varphi})\) to be the WOT-closure of all noncommutative polynomials in \(R_{Z_{1}},\ldots,R_{Z_{n}}\) and the identity. Now we can define the strict noncommutative variety

$$\mathcal{V}_{f,\varphi,\mathcal{I}}^{< }(\mathcal{H}):=\bigl\{ (X_{1},\ldots ,X_{n})\in\mathbb{D}_{f,\varphi}^{< }(\mathcal{H}): \omega(X_{1},\ldots,X_{n})=0 \mbox{ for any } \omega\in \mathcal{I}\bigr\} , $$

where \(\mathcal{I}\) is a WOT-closed two-sided ideal of the noncommutative Hardy algebra \(H^{\infty}(\mathbb{D}_{f,\varphi})\).

Noncommutative Poisson kernel

If \(T=(T_{1},\ldots,T_{n})\in\mathbb{D}_{f,\varphi}(\mathcal{H})\), we define the positive linear mapping

$$\varPhi_{f,\varphi,T}:B(\mathcal{H})\to B(\mathcal{H})\quad\mbox{by } \varPhi_{f,\varphi,T}(Y):=\sum_{|\alpha|\ge1}a_{\alpha}\bigl[\varphi (T)\bigr]_{\alpha}Y\bigl[\varphi(T)\bigr]_{\alpha}^{*}, $$

where the convergence is in the weak operator topology. We say that \(T=(T_{1},\ldots,T_{n})\) is a pure n-tuple of operators in \(\mathbb{D}_{f,\varphi}(\mathcal{H})\) if

$$\text{SOT-}\lim_{m\to\infty}\varPhi_{f,\varphi,T}^{m}(I)=0. $$

The set of all pure elements of \(\mathbb{D}_{f,\varphi}(\mathcal{H})\) is denoted by \(\mathbb{D}_{f,\varphi}^{\mathrm{pure}}(\mathcal{H})\). Notice that \((M_{Z_{1}},\ldots,M_{Z_{n}})\) is in \(\mathbb{D}_{f,\varphi }^{\mathrm{pure}}(\mathbb{H}_{f}^{2}(\varphi))\). Moreover, we refer to the n-tuple \((M_{Z_{1}},\ldots, M_{Z_{n}})\) as the universal model associated with the abstract noncommutative domain \(\mathbb{D}_{f,\varphi}\). An n-tuple \(T\in\mathbb {D}_{f,\varphi}(\mathcal{H})\) is called completely non-coisometric (c.n.c.) if there is no vector \(h\in\mathcal{H}\), \(h\neq0\), such that

$$\bigl\langle \varPhi_{f,\varphi,T}^{m}(I)h,h\bigr\rangle = \Vert h \Vert ^{2}\quad\mbox{for any } m=1,2,\ldots. $$

The set of all c.n.c. elements of \(\mathbb{D}_{f,\varphi}(\mathcal{H})\) is denoted by \(\mathbb{D}_{f,\varphi}^{\mathrm{cnc}}(\mathcal{H})\). Note that

$$\mathbb{D}_{f,\varphi}^{\mathrm{pure}}(\mathcal{H})\subseteq \mathbb{D}_{f,\varphi }^{\mathrm{cnc}}(\mathcal{H})\subseteq \mathbb{D}_{f,\varphi}(\mathcal{H}). $$

Similarly, we have

$$\mathcal{V}_{f,\varphi,\mathcal{I}}^{\mathrm{pure}}(\mathcal{H})\subseteq\mathcal {V}_{f,\varphi,\mathcal{I}}^{\mathrm{cnc}}(\mathcal{H})\subseteq\mathcal {V}_{f,\varphi,\mathcal{I}}(\mathcal{H}). $$

Moreover, it is obvious that the n-tuple \((B_{1},\ldots,B_{n})\) is in the noncommutative variety \(\mathcal{V}_{f,\varphi,\mathcal {I}}^{\mathrm{pure}}(\mathcal{N}_{f,\varphi,\mathcal{I}})\), where \(B_{i}:=P_{\mathcal{N}_{f,\varphi,\mathcal{I}}}M_{Z_{i}}|_{\mathcal {N}_{f,\varphi,\mathcal{I}}}\) for \(i=1,\ldots,n\). We refer to the n-tuple \((B_{1},\ldots,B_{n})\) as the universal model associated with the abstract noncommutative variety \(\mathcal{V}_{f,\varphi,\mathcal{I}}\).

We define the noncommutative Poisson kernel associated with the n-tuple \(T:=(T_{1},\ldots,T_{n})\in\mathbb{D}_{f,\varphi}(\mathcal{H})\) to be the operator \(K_{f,\varphi,T}:\mathcal{H}\to\mathbb{H}_{f}^{2}(\varphi)\otimes\overline {\Delta_{f,\varphi,T}(\mathcal{H})}\) defined by

$$K_{f,\varphi,T}h:=\sum_{\alpha\in\mathbb{F}_{n}^{+}}b_{\alpha}\varphi_{\alpha}\otimes\Delta_{f,\varphi,T}\bigl[\varphi(T) \bigr]_{\alpha}^{*}h,\quad h\in\mathcal{H}, $$

where \(\Delta_{f,\varphi,T}:=(I-\varPhi_{f,\varphi,T}(I))^{\frac{1}{2}}\) and the coefficients \(b_{\alpha}\), \(\alpha\in\mathbb{F}_{n}^{+}\), are given by relation (2.1).

Characteristic function

We consider the full Fock space of \(H_{n}\) defined by

$${F^{2}(H_{n}):=\mathbb{C}1\oplus\bigoplus _{m\geq1}H_{n}^{\otimes m},} $$

where \(H_{n}^{\otimes m}\) is the Hilbert tensor product of m copies of \(H_{n}\). Define the left creation operators \(S_{i}\), \(i = 1,\ldots,n\), acting on \(F^{2}(H_{n})\) by setting \(S_{i}\xi:= e_{i}\otimes\xi\), \(\xi\in F^{2}(H^{n})\). If \(A \in B(F^{2}(H_{n})\otimes\mathcal{G} ,F^{2}(H_{n})\otimes\mathcal{K} )\) and

$${\bigl(S_{i}^{*}\otimes I_{\mathcal{K}} \bigr)A(S_{j} \otimes I_{\mathcal{G}}) = \delta _{ij}A,\quad i,j= 1, \ldots,n,} $$

then A is called multi-Toeplitz with respect to \(S_{1},\ldots,S_{n}\). Moreover, if \(A \in B(F^{2}(H_{n})\otimes\mathcal{G} , F^{2}(H_{n})\otimes\mathcal{K} )\) and

$${A(S_{i}\otimes I_{\mathcal{G}})=(S_{i}\otimes I_{\mathcal{K}})A,\quad i= 1,\ldots,n,} $$

then A is called multi-analytic with respect to \(S_{1},\ldots,S_{n}\) (see [17, 19]). We remark that several results concerning the full Fock space \(F^{2}(H_{n})\) have been extended to the Hilbert space \(\mathbb{H}^{2}_{f}(\varphi)\) (see [25, 26, 28]). If \(A\in B(\mathbb{H}_{f}^{2}(\varphi)\otimes\mathcal{G}, \mathbb {H}_{f}^{2}(\varphi)\otimes\mathcal{K})\), and

$${A(M_{Z_{i}}\otimes I_{\mathcal{G}})=(M_{Z_{i}}\otimes I_{\mathcal{K}})A,\quad i= 1,\ldots,n,} $$

then A is called multi-analytic with respect to \(M_{Z_{1}},\ldots ,M_{Z_{n}}\) (see Definition 3.1 of [28]). Indeed, this definition is an analogy.

Let \(f=\sum_{|\alpha|\ge1}a_{\alpha}X_{\alpha}\) be a positive regular free holomorphic function and define the set \(\varGamma:=\{\alpha\in\mathbb {F}_{n}^{+}:a_{\alpha}\neq0\}\) and \(N:=\operatorname{card}(\varGamma)\). If \(\varphi=(\varphi_{1},\ldots ,\varphi_{n})\) is an n-tuple of formal power series with the model property and \(T:=(T_{1},\ldots,T_{n})\in \mathbb{D}_{f,\varphi}(\mathcal{H})\), we define the row operator

$$C_{f,\varphi,T}:=\bigl[\sqrt{a_{\widetilde{\alpha}}}\bigl[\varphi(T) \bigr]_{\widetilde {\alpha}}:\alpha\in\varGamma\bigr], $$

where the entries are arranged in the lexicographic order of \(\varGamma \subset\mathbb{F}_{n}^{+}\), and α̃ is the reverse of \(\alpha=g_{i_{1}}\cdots g_{i_{k}}\), i.e., \(\widetilde{\alpha}=g_{i_{k}}\cdots g_{i_{1}}\). Note that \(C_{f,\varphi,T}\) is an operator acting from \(\mathcal {H}^{(N)}\) (the completion of the direct sum of N copies of \(\mathcal {H}\)) to \(\mathcal{H}\).

Let \((M_{Z_{1}},\ldots,M_{Z_{n}})\) be the universal model associated with the abstract noncommutative domain \(\mathbb{D}_{f,\varphi}\). We introduce the characteristic function of an n-tuple \(T:=(T_{1},\ldots,T_{n})\in \mathbb{D}_{f,\varphi}(\mathcal{H})\) to be the multi-analytic operator with respect to \(M_{Z_{1}},\ldots,M_{Z_{n}}\),

$$\varTheta_{f,\varphi,T}:\mathbb{H}_{f}^{2}(\varphi) \otimes\mathcal {D}_{C_{f,\varphi,T}^{*}}\to\mathbb{H}_{f}^{2}( \varphi)\otimes\mathcal {D}_{C_{f,\varphi,T}} $$

with formal Fourier representation

$$\begin{aligned} &{-}I\otimes C_{f,\varphi,T}+(I\otimes\Delta_{C_{f,\varphi,T}})\biggl(I-\sum _{|\alpha|\ge1}a_{\widetilde{\alpha}}R_{\varphi_{\alpha}}\otimes \bigl[\varphi (T)\bigr]_{\widetilde{\alpha}}^{*}\biggr)^{-1} \\ &\quad\times [\sqrt{a_{\widetilde{\alpha}}}R_{\varphi_{\alpha}}\otimes I:\alpha\in \varGamma](I\otimes\Delta_{C_{f,\varphi,T}^{*}}), \end{aligned}$$

where \(R_{\varphi_{1}},\ldots,R_{\varphi_{n}}\) are the right multiplication operators by the formal power series \(\varphi_{1},\ldots,\varphi_{n}\), respectively, on the Hilbert space \(\mathbb{H}_{f}^{2}(\varphi)\). The defect operators associated with the row contraction \(C_{f,\varphi,T}\) are

$$\begin{aligned}& \Delta_{C_{f,\varphi,T}} :=\bigl(I-C_{f,\varphi,T}C_{f,\varphi,T}^{*} \bigr)^{\frac {1}{2}}\in B(\mathcal{H}), \\& \Delta_{C_{f,\varphi,T}^{*}} :=\bigl(I-C_{f,\varphi,T}^{*}C_{f,\varphi ,T} \bigr)^{\frac{1}{2}}\in B\bigl(\mathcal{H}^{(N)}\bigr), \end{aligned}$$

and the defect spaces are \(\mathcal{D}_{C_{f,\varphi,T}}:=\overline {\Delta_{C_{f,\varphi,T}} \mathcal{H}}\) and \(\mathcal{D}_{C_{f,\varphi,T}^{*}}:= \overline{\Delta_{C_{f,\varphi,T}^{*}} \mathcal{H}^{(N)}}\).

Constrained characteristic functions

In this section, we present a functional model theorem for completely non-coisometric n-tuples of operators in the noncommutative variety \(\mathcal{V}_{f,\varphi,\mathcal{I}}(\mathcal{H})\) in terms of constrained characteristic functions. Moreover, we prove that the constrained characteristic function is a complete unitary invariant for this class of elements. Indeed, this result can be viewed as the noncommutative analogue of the classical Sz.-Nagy–Foiaş functional model for completely nonunitary contractions.

Let \(T=(T_{1},\ldots,T_{n})\) be an n-tuple of operators in \(\mathcal {V}_{f,\varphi,\mathcal{I}}^{\mathrm{cnc}}(\mathcal{H})\). The constrained Poisson kernel is the operator \(K_{f,\varphi,T}^{(\mathcal{I})}:\mathcal{H}\to\mathcal {N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{D}_{C_{f,\varphi,T}}\) defined by

$$K_{f,\varphi,T}^{(\mathcal{I})}:=(P_{\mathcal{N}_{f,\varphi,\mathcal {I}}}\otimes I_{\mathcal{D}_{C_{f,\varphi,T}}})K_{f,\varphi,T}, $$

where \(K_{f,\varphi,T}\) is the noncommutative Poisson kernel associated with f, φ, and T.

First, we present some basic properties for the constrained Poisson kernel \(K_{f,\varphi,T}^{(\mathcal{I})}\) associated with f, φ, T, and \(\mathcal{I}\).

Theorem 3.1

Let\(f:=\sum_{\alpha\in\mathbb{F}_{n}^{+}}a_{\alpha}Z_{\alpha}\)be a positive regular free holomorphic function, and let\(\varphi=(\varphi_{1},\ldots ,\varphi_{n})\)be ann-tuple of formal power series with model property. Let\(\mathcal{I} \neq H^{\infty}(\mathbb{D}_{f,\varphi})\)be a WOT-closed two-sided ideal of the noncommutative Hardy algebra\(H^{\infty}(\mathbb{D}_{f,\varphi})\). If\(T=(T_{1},\ldots,T_{n})\)is ann-tuple of operators in\(\mathcal{V}_{f,\varphi,\mathcal {I}}^{\mathrm{cnc}}(\mathcal{H})\), then the following statements hold:

  1. (i)

    \(K_{f,\varphi,T}^{(\mathcal{I})}T_{i}^{*}=(B_{i}^{*}\otimes I_{\mathcal{D}_{C_{f,\varphi,T}}})K_{f,\varphi,T}^{(\mathcal{I})}\), \(i=1,\ldots,n\);

  2. (ii)

    \(K_{f,\varphi,T}^{(\mathcal{I})}\)is an isometry if and only ifTis pure,

where\(K_{f,\varphi,T}^{(\mathcal{I})}\)is the constrained Poisson kernel associated withf, φ, T, and\(\mathcal{I}\).

Proof

(i) According to the proof of Theorem 2.1 from [28], we know that

$$K_{f,\varphi,T}T_{i}^{*}=\bigl(M_{Z_{i}}^{*}\otimes I_{\mathcal{D}_{C_{f,\varphi ,T}}}\bigr)K_{f,\varphi,T},\quad i=1,\ldots,n, $$

where \(K_{f,\varphi,T}\) is the noncommutative Poisson kernel associated with f, φ, and T. Hence, we have

$$ K_{f,\varphi,T}^{*}\bigl(p(M_{Z_{1}}, \ldots,M_{Z_{n}})\otimes I_{\mathcal {D}_{C_{f,\varphi,T}}}\bigr)=p(T_{1}, \ldots,T_{n})K_{f,\varphi,T}^{*} $$
(3.1)

for any polynomial p in \(M_{Z_{1}},\ldots,M_{Z_{n}}\). Assume that

$$\phi(V_{1},\ldots,V_{n})=\sum _{k=0}^{\infty}\sum_{|\alpha|=k}d_{\alpha}V_{\alpha},\quad d_{\alpha}\in\mathbb{C}, $$

is an element in the noncommutative Hardy algebra \(F^{\infty}(\mathcal {D}_{f})\). Then we deduce that

$$\phi(rV_{1},\ldots,rV_{n})=\sum _{k=0}^{\infty}\sum_{|\alpha|=k}r^{|\alpha |}d_{\alpha}V_{\alpha}\quad\mbox{for any } 0< r< 1 $$

is in the noncommutative domain algebra \(\mathcal{A}(\mathcal{D}_{f})\). Moreover, since φ has model property, we have

$$M_{\varphi_{i}}=\varphi_{i}(M_{Z_{1}}, \ldots,M_{Z_{n}}),\quad i=1,\ldots,n, $$

where \((M_{Z_{1}},\ldots,M_{Z_{n}})\) is either in the set \(\mathcal {C}_{\varphi}^{\mathrm{SOT}}(\mathbb{H}_{f}^{2}(\varphi))\) or \(\mathcal{C}_{\varphi}^{\mathrm{rad}}(\mathbb{H}_{f}^{2}(\varphi))\). Using (2.3), we conclude that

$$V_{i}=U\varphi_{i}(M_{Z_{1}}, \ldots,M_{Z_{n}})U^{-1},\quad i=1,\ldots,n. $$

Therefore, we obtain

$$\phi\bigl(r\varphi_{1}(M_{Z}),\ldots,r \varphi_{n}(M_{Z})\bigr)=\sum _{k=0}^{\infty}\sum_{|\alpha|=k}r^{|\alpha|}d_{\alpha}\bigl[\varphi(M_{Z})\bigr]_{\alpha}, $$

where the series is convergent in the operator norm topology. Hence, due to (3.1), we infer that

$$K_{f,\varphi,T}^{*}\bigl[\phi\bigl(r\varphi_{1}(M_{Z}), \ldots,r\varphi_{n}(M_{Z})\bigr)\otimes I_{\mathcal{D}_{C_{f,\varphi,T}}}\bigr]=\phi\bigl(r\varphi_{1}(T),\ldots,r\varphi _{n}(T)\bigr)K_{f,\varphi,T}^{*} $$

for any \(\phi(V_{1},\ldots,V_{n})\in F^{\infty}(\mathcal{D}_{f})\) and \(0< r<1\). Since \(T=(T_{1},\ldots,T_{n})\) is in \(\mathbb{D}_{f,\varphi}^{\mathrm{cnc}}(\mathcal {H})\) and \(M_{Z}=(M_{Z_{1}},\ldots,M_{Z_{n}})\) is in \(\mathbb{D}_{f,\varphi }^{\mathrm{pure}}(\mathbb{H}_{f}^{2}(\varphi))\), we deduce that \(\varphi(T)=(\varphi _{1}(T),\ldots,\varphi_{n}(T))\) is a completely non-coisometric n-tuple of operators in the noncommutative domain \(\mathcal{D}_{f}(\mathcal{H})\) and \(\varphi (M_{Z})=(\varphi_{1}(M_{Z}),\ldots,\varphi_{n}(M_{Z}))\) is a pure n-tuple of operators in \(\mathcal{D}_{f}(\mathbb{H}_{f}^{2}(\varphi))\). Taking into account that

$$\bigl\Vert \phi\bigl(r\varphi_{1}(M_{Z}),\ldots,r \varphi_{n}(M_{Z})\bigr) \bigr\Vert \le \bigl\Vert \phi(V_{1},\ldots ,V_{n}) \bigr\Vert $$

and using \(F^{\infty}(\mathcal{D}_{f})\)-functional calculus (see [24]), we infer that

$$K_{f,\varphi,T}^{*}\bigl[\phi\bigl(\varphi_{1}(M_{Z}), \ldots,\varphi_{n}(M_{Z})\bigr)\otimes I_{\mathcal{D}_{C_{f,\varphi,T}}}\bigr]=\phi\bigl(\varphi_{1}(T),\ldots,\varphi _{n}(T)\bigr)K_{f,\varphi,T}^{*} $$

for any \(\phi(V_{1},\ldots,V_{n})\in F^{\infty}(\mathcal{D}_{f})\). Using Proposition 4.2 from [28], we know that if \(\theta\in H^{\infty}(\mathbb{D}_{f,\varphi})\), there is \(\chi=\sum_{\alpha\in\mathbb {F}_{n}^{+}}c_{\alpha}V_{\alpha}\) in \(F^{\infty}(\mathcal{D}_{f})\) such that

$$\theta=\text{SOT-}\lim_{r\to1}\sum _{k=0}^{\infty}\sum_{|\alpha |=k}c_{\alpha}r^{|\alpha|}\bigl[\varphi(M_{Z})\bigr]_{\alpha}=\chi \bigl(\varphi(M_{Z})\bigr). $$

Indeed, this implies that

$$H^{\infty}(\mathbb{D}_{f,\varphi})=\bigl\{ \chi\bigl( \varphi(M_{Z})\bigr):\chi\in F^{\infty}( \mathcal{D}_{f})\bigr\} . $$

Moreover, since \(T=(T_{1},\ldots,T_{n})\) is in \(\mathcal{V}_{f,\varphi ,\mathcal{I}}^{\mathrm{cnc}}(\mathcal{H})\), we deduce that \(\varphi(T)=(\varphi_{1}(T), \ldots,\varphi_{n}(T))\) is also a completely non-coisometric n-tuple of operators in \(\mathcal{D}_{f}(\mathcal{H})\). Using \(F^{\infty}(\mathcal{D}_{f})\)-functional calculus, we obtain that

$$\theta(T_{1},\ldots,T_{n})=\text{SOT-}\lim _{r\to1}\sum_{k=0}^{\infty}\sum_{|\alpha|=k}c_{\alpha}r^{|\alpha|} \bigl[\varphi(T)\bigr]_{\alpha}=\chi\bigl(\varphi _{1}(T), \ldots,\varphi_{n}(T)\bigr). $$

This shows that

$$ K_{f,\varphi,T}^{*}(\omega\otimes I_{\mathcal{D}_{C_{f,\varphi ,T}}})= \omega(T)K_{f,\varphi,T}^{*} $$
(3.2)

for any \(\omega\in H^{\infty}(\mathbb{D}_{f,\varphi})\). Consequently, we deduce that

$$\bigl\langle \bigl(\omega^{*}\otimes I_{\mathcal{D}_{C_{f,\varphi,T}}}\bigr)K_{f,\varphi ,T}h,1 \otimes d\bigr\rangle =\bigl\langle K_{f,\varphi,T}\omega(T)^{*}h,1\otimes d\bigr\rangle $$

for any \(\omega\in H^{\infty}(\mathbb{D}_{f,\varphi})\), \(h\in\mathcal {H}\), and \(d\in\mathcal{D}_{C_{f,\varphi,T}}\). Since \(\mathcal{I}\) is a WOT-closed two-sided ideal of \(H^{\infty}(\mathbb{D}_{f,\varphi})\), we have

$$\mathcal{M}_{f,\varphi,\mathcal{I}}=\overline{\mathcal{I}(1)}. $$

Note that \(T\in\mathcal{V}_{f,\varphi,\mathcal{I}}^{\mathrm{cnc}}(\mathcal {H})\). Then we obtain

$$\bigl\langle K_{f,\varphi,T}h,\omega(1)\otimes d\bigr\rangle =0 $$

for any \(\omega\in\mathcal{I}\), \(h\in\mathcal{H}\), and \(d\in\mathcal {D}_{C_{f,\varphi,T}}\). Therefore, we conclude that

$$K_{f,\varphi,T}(\mathcal{H})\subseteq\mathcal{N}_{f,\varphi,\mathcal {I}}\otimes \mathcal{D}_{C_{f,\varphi,T}}, $$

which implies that

$$ K_{f,\varphi,T}^{(\mathcal{I})}h=(P_{\mathcal{N}_{f,\varphi,\mathcal {I}}}\otimes I_{\mathcal{D}_{C_{f,\varphi,T}}})K_{f,\varphi ,T}h=K_{f,\varphi,T}h,\quad h\in \mathcal{H}. $$
(3.3)

On the other hand, since \(\mathcal{N}_{f,\varphi,\mathcal{I}}\) is an invariant subspace under \(M_{Z_{1}}^{*},\ldots,M_{Z_{n}}^{*}\), we have

$$B_{\alpha}=P_{\mathcal{N}_{f,\varphi,\mathcal{I}}}M_{Z_{\alpha}}|_{\mathcal {N}_{f,\varphi,\mathcal{I}}}\quad \mbox{for any } \alpha\in\mathbb{F}_{n}^{+}. $$

According to Proposition 4.2 of [28], we know that, for any \(\nu\in H^{\infty}(\mathbb{D}_{f,\varphi})\), there exists \(\chi\in F^{\infty}(\mathcal{D}_{f})\) such that

$$\begin{aligned} \nu(M_{Z_{1}},\ldots,M_{Z_{n}})&= \chi\bigl( \varphi_{1}(M_{Z}),\ldots,\varphi _{n}(M_{Z}) \bigr) \\ &= \text{SOT-}\lim_{r\to1}\chi\bigl(r \varphi_{1}(M_{Z}),\ldots,r\varphi_{n}(M_{Z}) \bigr). \end{aligned}$$

Since \((B_{1},\ldots,B_{n})\) is in the noncommutative variety \(\mathcal {V}_{f,\varphi,\mathcal{I}}^{\mathrm{pure}}(\mathcal{N}_{f,\varphi,\mathcal {I}})\), we obtain that \((\varphi_{1}(B),\ldots, \varphi_{n}(B))\) is a pure n-tuple of operators in \(\mathcal{D}_{f}(\mathcal{N}_{f,\varphi,\mathcal{I}})\). Consequently, using \(F^{\infty}(\mathcal{D}_{f})\)-functional calculus, we deduce that

$$ \nu(B_{1},\ldots,B_{n})=P_{\mathcal{N}_{f,\varphi,\mathcal{I}}} \nu (M_{Z_{1}},\ldots,M_{Z_{n}})|_{\mathcal{N}_{f,\varphi,\mathcal{I}}} $$
(3.4)

for any \(\nu\in H^{\infty}(\mathbb{D}_{f,\varphi})\). Applying (3.2), (3.3), and (3.4), we infer that

$$\begin{aligned} K_{f,\varphi,T}^{(\mathcal{I})}\nu(T_{1},\ldots,T_{n})^{*}={}& (P_{\mathcal {N}_{f,\varphi,\mathcal{I}}}\otimes I_{\mathcal{D}_{C_{f,\varphi ,T}}})\bigl[\nu(M_{Z_{1}}, \ldots,M_{Z_{n}})^{*}\otimes I_{\mathcal{D}_{C_{f,\varphi ,T}}}\bigr] \\ & \times(P_{\mathcal{N}_{f,\varphi,\mathcal{I}}}\otimes I_{\mathcal {D}_{C_{f,\varphi,T}}})K_{f,\varphi,T} \\ ={}& \bigl[\nu(B_{1},\ldots,B_{n})^{*}\otimes I_{\mathcal{D}_{C_{f,\varphi ,T}}}\bigr]K_{f,\varphi,T}^{(\mathcal{I})} \end{aligned}$$

for any \(\nu(B_{1},\ldots,B_{n})\in H^{\infty}(\mathcal{V}_{f,\varphi,\mathcal {I}})\). In particular, we have

$$K_{f,\varphi,T}^{(\mathcal{I})}T_{i}^{*}=\bigl(B_{i}^{*} \otimes I_{\mathcal {D}_{C_{f,\varphi,T}}}\bigr)K_{f,\varphi,T}^{(\mathcal{I})},\quad i=1, \ldots,n. $$

(ii) Due to (3.3), we obtain

$$\begin{aligned} \bigl\langle \bigl(K_{f,\varphi,T}^{(\mathcal{I})}\bigr)^{*}K_{f,\varphi,T}^{(\mathcal {I})}h,h \bigr\rangle &= \Vert K_{f,\varphi,T}h \Vert ^{2} \\ &= \Vert h \Vert ^{2}-\lim_{m\to\infty}\bigl\langle \varPhi_{f,\varphi ,T}^{m}{(I)}h,h\bigr\rangle . \end{aligned}$$

Hence, we deduce that

$$ \bigl(K_{f,\varphi,T}^{(\mathcal{I})}\bigr)^{*}K_{f,\varphi,T}^{(\mathcal {I})}=I- \varPhi_{f,\varphi,T}^{\infty}(I), $$
(3.5)

where \(\varPhi_{f,\varphi,T}^{\infty}(I):=\text{SOT-}\lim_{m\to\infty }\varPhi_{f,\varphi,T}^{m}(I)\). Therefore, (ii) holds. This completes the proof. □

We define the constrained characteristic function associated with an n-tuple \(T:=(T_{1},\ldots,T_{n})\in\mathcal{V}_{f,\varphi,\mathcal {I}}^{\mathrm{cnc}}(\mathcal{H})\) to be the multi-analytic operator with respect to the constrained weighted shifts \(B_{1},\ldots,B_{n}\),

$$\varTheta_{f,\varphi,T}^{(\mathcal{I})}:\mathcal{N}_{f,\varphi,\mathcal {I}}\otimes \mathcal{D}_{C_{f,\varphi,T}^{*}}\to\mathcal{N}_{f,\varphi ,\mathcal{I}}\otimes \mathcal{D}_{C_{f,\varphi,T}}, $$

with the formal Fourier representation

$$\begin{aligned} &{-}I_{\mathcal{N}_{f,\varphi,\mathcal{I}}}\otimes C_{f,\varphi ,T}+(I_{\mathcal{N}_{f,\varphi,\mathcal{I}}}\otimes \Delta_{C_{f,\varphi,T}}) \biggl(I_{\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{H}}-\sum _{|\alpha|\ge1}a_{\widetilde{\alpha}}D_{\alpha}\otimes\bigl[ \varphi (T)\bigr]_{\widetilde{\alpha}}^{*}\biggr)^{-1} \\ &\quad\times [\sqrt{a_{\widetilde{\alpha}}}D_{\alpha}\otimes I_{\mathcal{H}}:\alpha\in\varGamma](I_{\mathcal{N}_{f,\varphi,\mathcal{I}}}\otimes \Delta_{C_{f,\varphi,T}^{*}}), \end{aligned}$$

where \(D_{i}=P_{\mathcal{N}_{f,\varphi,\mathcal{I}}}R_{\varphi _{i}}|_{\mathcal{N}_{f,\varphi,\mathcal{I}}}\), \(i=1,\ldots,n\), and \(R_{\varphi_{1}}, \ldots,R_{\varphi_{n}}\) are the right multiplication operators by the power series \(\varphi_{1},\ldots,\varphi_{n}\), respectively, on the Hilbert space \(\mathbb{H}_{f}^{2}(\varphi)\).

We provide a factorization result for the constrained characteristic function, which will play an important role in our investigation.

Theorem 3.2

Let\(f:=\sum_{\alpha\in\mathbb{F}_{n}^{+}}a_{\alpha}Z_{\alpha}\)be a positive regular free holomorphic function, and let\(\varphi=(\varphi_{1},\ldots ,\varphi_{n})\)be ann-tuple of formal power series with model property. Let\(\mathcal{I} \neq H^{\infty}(\mathbb{D}_{f,\varphi})\)be a WOT-closed two-sided ideal of the noncommutative Hardy algebra\(H^{\infty}(\mathbb{D}_{f,\varphi})\). Then

$$I_{\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{D}_{C_{f,\varphi ,T}}}-\varTheta_{f,\varphi,T}^{(\mathcal{I})}\bigl( \varTheta_{f,\varphi ,T}^{(\mathcal{I})}\bigr)^{*}= K_{f,\varphi,T}^{(\mathcal{I})} \bigl(K_{f,\varphi,T}^{(\mathcal{I})}\bigr)^{*}, $$

where\(\varTheta_{f,\varphi,T}^{(\mathcal{I})}\)is the constrained characteristic function and\(K_{f,\varphi,T}^{(\mathcal{I})}\)is the corresponding constrained Poisson kernel.

Proof

Due to Theorem 6.1 of [28], we know that

$$I_{\mathbb{H}_{f}^{2}(\varphi)\otimes\mathcal{D}_{C_{f,\varphi ,T}}}-\varTheta_{f,\varphi,T}\varTheta_{f,\varphi,T}^{*}= K_{f,\varphi,T}K_{f,\varphi,T}^{*}. $$

According to the proof of Theorem 3.1, we have

$$K_{f,\varphi,T}(\mathcal{H})\subseteq\mathcal{N}_{f,\varphi,\mathcal {I}}\otimes \mathcal{D}_{C_{f,\varphi,T}}\subseteq\mathbb{H}_{f}^{2}( \varphi )\otimes\mathcal{D}_{C_{f,\varphi,T}}. $$

Hence, we infer that

$$\begin{aligned} & I_{\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal {D}_{C_{f,\varphi,T}}}-P_{\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes \mathcal{D}_{C_{f,\varphi,T}}} \varTheta_{f,\varphi,T} \varTheta_{f,\varphi,T}^{*}|_{\mathcal {N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{D}_{C_{f,\varphi,T}}} \\ &\quad= P_{\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal {D}_{C_{f,\varphi,T}}} K_{f,\varphi,T}K_{f,\varphi,T}^{*}|_{\mathcal{N}_{f,\varphi,\mathcal {I}}\otimes\mathcal{D}_{C_{f,\varphi,T}}}. \end{aligned}$$
(3.6)

Since \(\mathcal{N}_{f,\varphi,\mathcal{I}}\) is an invariant subspace under \(R_{\varphi_{1}}^{*},\ldots,R_{\varphi_{n}}^{*}\), we obtain

$$ \varTheta_{f,\varphi,T}^{*}(\mathcal{N}_{f,\varphi,\mathcal{I}} \otimes \mathcal{D}_{C_{f,\varphi,T}})\subseteq\mathcal{N}_{f,\varphi,\mathcal {I}}\otimes \mathcal{D}_{C_{f,\varphi,T}^{*}} $$
(3.7)

and

$$ P_{\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{D}_{C_{f,\varphi ,T}}}\varTheta_{f,\varphi,T}|_{\mathcal{N}_{f,\varphi,\mathcal {I}}\otimes\mathcal{D}_{C_{f,\varphi,T}^{*}}} =\varTheta_{f,\varphi,T}^{(\mathcal{I})}. $$
(3.8)

Applying (3.6), (3.7), and (3.8), we deduce that

$$I_{\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{D}_{C_{f,\varphi ,T}}}-\varTheta_{f,\varphi,T}^{(\mathcal{I})}\bigl( \varTheta_{f,\varphi ,T}^{(\mathcal{I})}\bigr)^{*}= K_{f,\varphi,T}^{(\mathcal{I})} \bigl(K_{f,\varphi,T}^{(\mathcal{I})}\bigr)^{*}. $$

This completes the proof. □

If \(A\in B(\mathbb{H}_{f}^{2}(\varphi)\otimes\mathcal{G},\mathbb {H}_{f}^{2}(\varphi)\otimes\mathcal{K})\) is a multi-analytic operator and A is a partial isometry, then we call it inner multi-analytic.

In what follows, we present a functional model theorem for completely non-coisometric n-tuples of operators in the noncommutative variety \(\mathcal{V}_{f,\varphi,\mathcal{I}}(\mathcal{H})\) in terms of constrained characteristic functions.

Theorem 3.3

Let\(f:=\sum_{\alpha\in\mathbb{F}_{n}^{+}}a_{\alpha}Z_{\alpha}\)be a positive regular free holomorphic function, and let\(\varphi=(\varphi_{1},\ldots ,\varphi_{n})\)be ann-tuple of formal power series with model property. Let\(\mathcal{I} \neq H^{\infty}(\mathbb{D}_{f,\varphi})\)be a WOT-closed two-sided ideal of the noncommutative Hardy algebra\(H^{\infty}(\mathbb{D}_{f,\varphi})\). If\(T:=(T_{1},\ldots,T_{n})\)is in the noncommutative variety\(\mathcal{V}_{f,\varphi,\mathcal {I}}^{\mathrm{cnc}}(\mathcal{H})\), then the following statements hold:

  1. (i)

    Tis unitarily equivalent to then-tuple\(\widetilde {T}:=(\widetilde{T}_{1},\ldots,\widetilde{T}_{n})\in\mathcal{V}_{f,\varphi ,\mathcal{I}}^{\mathrm{cnc}}(\widetilde{\mathcal{H}})\)on the Hilbert space

    $$\begin{aligned} \widetilde{\mathcal{H}}:={} & \bigl[(\mathcal{N}_{f,\varphi,\mathcal {I}}\otimes \mathcal{D}_{C_{f,\varphi,T}})\oplus\overline{\Delta _{\varTheta_{f,\varphi,T}^{(\mathcal{I})}}( \mathcal{N}_{f,\varphi ,\mathcal{I}}\otimes \mathcal{D}_{C_{f,\varphi,T}^{*}})}\bigr] \\ & \ominus\bigl\{ \varTheta_{f,\varphi,T}^{(\mathcal{I})}x\oplus\Delta _{\varTheta_{f,\varphi,T}^{(\mathcal{I})}}x:x\in\mathcal{N}_{f,\varphi ,\mathcal{I}}\otimes \mathcal{D}_{C_{f,\varphi,T}^{*}} \bigr\} , \end{aligned}$$

    where\(\Delta_{\varTheta_{f,\varphi,T}^{(\mathcal{I})}}=(I-(\varTheta _{f,\varphi,T}^{(\mathcal{I})})^{*}\varTheta_{f,\varphi,T}^{(\mathcal {I})})^{\frac{1}{2}}\)and each operator\(\widetilde{T}_{i}\), \(i=1,\ldots,n\), is uniquely defined by the relation

    $$\begin{aligned} & (P_{\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes \mathcal {D}_{C_{f,\varphi,T}}}|_{\widetilde{\mathcal{H}}})\widetilde{T}_{i}^{*}z \\ &\quad= \bigl(B_{i}^{*}\otimes I_{\mathcal{D}_{C_{f,\varphi,T}}}\bigr) (P_{\mathcal {N}_{f,\varphi,\mathcal{I}}\otimes \mathcal{D}_{C_{f,\varphi ,T}}}|_{\widetilde{\mathcal{H}}})z,\quad z\in\widetilde{\mathcal{H}}, \end{aligned}$$

    where\(P_{\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes \mathcal {D}_{C_{f,\varphi,T}}}|_{\widetilde{\mathcal{H}}}\)is an injective operator, \(P_{\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes \mathcal {D}_{C_{f,\varphi,T}}}\)is the orthogonal projection from the Hilbert space

    $$\widetilde{\mathcal{K}}:=(\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes \mathcal{D}_{C_{f,\varphi,T}})\oplus\overline{\Delta_{\varTheta _{f,\varphi,T}^{(\mathcal{I})}}( \mathcal{N}_{f,\varphi,\mathcal {I}}\otimes \mathcal{D}_{C_{f,\varphi,T}^{*}})} $$

    onto the subspace\(\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal {D}_{C_{f,\varphi,T}}\), and\(B_{i}=P_{\mathcal{N}_{f,\varphi,\mathcal{I}}}M_{Z_{i}}|_{\mathcal {N}_{f,\varphi,\mathcal{I}}}\)for any\(i=1,\ldots,n\);

  2. (ii)

    Tis in the noncommutative variety\(\mathcal{V}_{f,\varphi ,\mathcal{I}}^{\mathrm{pure}}(\mathcal{H})\)if and only if the constrained characteristic function\(\varTheta_{f,\varphi,T}^{(\mathcal{I})}\)is an inner multi-analytic operator. In this case, Tis unitarily equivalent to then-tuple

    $$\bigl(P_{\widetilde{\mathcal{H}}}(B_{1}\otimes I_{\mathcal{D}_{C_{f,\varphi ,T}}})|_{\widetilde{\mathcal{H}}}, \ldots,P_{\widetilde{\mathcal {H}}}(B_{n}\otimes I_{\mathcal{D}_{C_{f,\varphi,T}}})|_{\widetilde {\mathcal{H}}} \bigr), $$

    where\(P_{\widetilde{\mathcal{H}}}\)is the orthogonal projection from\(\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes \mathcal{D}_{C_{f,\varphi,T}}\)onto the Hilbert space\(\widetilde{\mathcal{H}}:=(\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes \mathcal{D}_{C_{f,\varphi,T}})\ominus\varTheta_{f,\varphi,T}^{(\mathcal {I})}(\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal {D}_{C_{f,\varphi,T}^{*}})\).

Proof

(i) We define the operator \(\varPsi:\mathcal{N}_{f,\varphi,\mathcal {I}}\otimes\mathcal{D}_{C_{f,\varphi,T}^{*}}\to\widetilde{\mathcal{K}}\) by setting

$$\varPsi x:=\varTheta_{f,\varphi,T}^{(\mathcal{I})}x\oplus\Delta _{\varTheta_{f,\varphi,T}^{(\mathcal{I})}}x, \quad x\in\mathcal {N}_{f,\varphi,\mathcal{I}}\otimes \mathcal{D}_{C_{f,\varphi,T}^{*}}. $$

It is obvious that Ψ is an isometry and

$$ \varPsi^{*}(y\oplus0)=\bigl(\varTheta_{f,\varphi,T}^{(\mathcal {I})} \bigr)^{*}y,\quad y\in\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal {D}_{C_{f,\varphi,T}}. $$
(3.9)

Hence, we infer that

$$\begin{aligned} \Vert y \Vert ^{2}&= \bigl\Vert P_{\widetilde{\mathcal{H}}}(y\oplus0) \bigr\Vert ^{2}+ \bigl\Vert \varPsi \varPsi ^{*}(y\oplus0) \bigr\Vert ^{2} \\ &= \bigl\Vert P_{\widetilde{\mathcal{H}}}(y\oplus0) \bigr\Vert ^{2}+ \bigl\Vert \bigl(\varTheta_{f,\varphi ,T}^{(\mathcal{I})}\bigr)^{*}y \bigr\Vert ^{2} \end{aligned}$$
(3.10)

for any \(y\in\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes \mathcal {D}_{C_{f,\varphi,T}}\), where \(P_{\widetilde{\mathcal{H}}}\) denotes the orthogonal projection from \(\widetilde{\mathcal{K}}\) onto \(\widetilde{\mathcal{H}}\). According to Theorem 3.2, we have

$$ \bigl\Vert \bigl(K_{f,\varphi,T}^{(\mathcal{I})}\bigr)^{*}y \bigr\Vert ^{2}+ \bigl\Vert \bigl(\varTheta_{f,\varphi ,T}^{(\mathcal{I})} \bigr)^{*}y \bigr\Vert ^{2}= \Vert y \Vert ^{2},\quad y\in\mathcal{N}_{f,\varphi ,\mathcal{I}}\otimes\mathcal{D}_{C_{f,\varphi,T}}. $$
(3.11)

Therefore, using (3.10) and (3.11), we deduce that

$$ \bigl\Vert \bigl(K_{f,\varphi,T}^{(\mathcal{I})}\bigr)^{*}y \bigr\Vert = \bigl\Vert P_{\widetilde{\mathcal {H}}}(y\oplus0) \bigr\Vert ,\quad y\in \mathcal{N}_{f,\varphi,\mathcal{I}}\otimes \mathcal{D}_{C_{f,\varphi,T}}. $$
(3.12)

On the other hand, due to (3.3), we obtain

$$\bigl\Vert K_{f,\varphi,T}^{(\mathcal{I})}h \bigr\Vert ^{2}= \Vert h \Vert ^{2}-\lim_{m\to\infty}\bigl\langle \varPhi_{f,\varphi,T}^{m}(I)h,h\bigr\rangle , \quad h\in\mathcal{H}. $$

Hence, if \(K_{f,\varphi,T}^{(\mathcal{I})}h=0\), then we have

$$\Vert h \Vert ^{2}=\lim_{m\to\infty}\bigl\langle \varPhi_{f,\varphi,T}^{m}(I)h,h\bigr\rangle . $$

Since T is in \(\mathcal{V}_{f,\varphi,\mathcal{I}}^{\mathrm{cnc}}(\mathcal {H})\), we infer that \(h=0\), which implies that \(K_{f,\varphi ,T}^{(\mathcal{I})}\) is an injective operator and range \((K_{f,\varphi,T}^{(\mathcal{I})})^{*}\) is dense in \(\mathcal{H}\).

Let \(z\in\widetilde{\mathcal{H}}\) and assume that \(z\perp P_{\widetilde {\mathcal{H}}}(y\oplus0)\) for any \(y\in\mathcal{N}_{f,\varphi,\mathcal {I}}\otimes\mathcal{D}_{C_{f,\varphi,T}}\). Taking into account that

$$\widetilde{\mathcal{K}}=\{y\oplus0:y\in\mathcal{N}_{f,\varphi,\mathcal {I}}\otimes \mathcal{D}_{C_{f,\varphi,T}}\}\vee\bigl\{ \varTheta_{f,\varphi ,T}^{(\mathcal{I})}x \oplus\Delta_{\varTheta_{f,\varphi,T}^{(\mathcal{I})}}x: x\in\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal {D}_{C_{f,\varphi,T}^{*}}\bigr\} . $$

Consequently, we obtain \(z=0\). This shows that

$$ \widetilde{\mathcal{H}}=\bigl\{ P_{\widetilde{\mathcal{H}}}(y\oplus0):y \in \mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{D}_{C_{f,\varphi ,T}}\bigr\} ^{-}. $$
(3.13)

Applying (3.12) and (3.13), we deduce that there exists a unique unitary operator \(W:\mathcal{H}\to\widetilde{\mathcal{H}}\) such that

$$W\bigl(K_{f,\varphi,T}^{(\mathcal{I})}y\bigr)=P_{\widetilde{\mathcal{H}}}(y\oplus 0), \quad y\in\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal {D}_{C_{f,\varphi,T}}. $$

Moreover, using (3.9) and Theorem 3.2 , we have

$$\begin{aligned} P_{\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{D}_{C_{f,\varphi ,T}}}W\bigl(K_{f,\varphi,T}^{(\mathcal{I})}\bigr)^{*}y&= P_{\mathcal{N}_{f,\varphi ,\mathcal{I}}\otimes\mathcal{D}_{C_{f,\varphi,T}}}P_{\widetilde {\mathcal{H}}}(y\oplus0) \\ &= y-P_{\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal {D}_{C_{f,\varphi,T}}}\varPsi\varPsi^{*}(y\oplus0) \\ &= y-\varTheta_{f,\varphi,T}^{(\mathcal{I})}\bigl(\varTheta_{f,\varphi ,T}^{(\mathcal{I})} \bigr)^{*}y \\ &= K_{f,\varphi,T}^{(\mathcal{I})}\bigl(K_{f,\varphi,T}^{(\mathcal{I})} \bigr)^{*}y \end{aligned}$$

for any \(y\in\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal {D}_{C_{f,\varphi,T}}\). Since the range \((K_{f,\varphi,T}^{(\mathcal {I})})^{*}\) is dense in \(\mathcal{H}\), we infer that

$$ P_{\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{D}_{C_{f,\varphi ,T}}}W=K_{f,\varphi,T}^{(\mathcal{I})}. $$
(3.14)

Let \(\widetilde{T}_{i}:\widetilde{\mathcal{H}}\to\widetilde{\mathcal{H}}\) be the transform of \(T_{i}\) under the unitary operator \(W:\mathcal{H}\to \widetilde{\mathcal{H}}\), i.e.,

$$\widetilde{T}_{i}=WT_{i}W^{*},\quad i=1,\ldots,n. $$

Since the constrained Poisson kernel \(K_{f,\varphi,T}^{(\mathcal{I})}\) is an injective operator, due to (3.14), we deduce that

$$P_{\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{D}_{C_{f,\varphi ,T}}}|_{\widetilde{\mathcal{H}}}=K_{f,\varphi,T}^{(\mathcal{I})}W^{*} $$

is an injective operator acting from \(\widetilde{\mathcal{H}}\) to \(\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{D}_{C_{f,\varphi,T}}\). Consequently, according to (3.14) and Theorem 3.1, we have

$$\begin{aligned} (P_{\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal {D}_{C_{f,\varphi,T}}}|_{\widetilde{\mathcal{H}}})\widetilde{T}_{i}^{*}Wh&= (P_{\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal {D}_{C_{f,\varphi,T}}}|_{\widetilde{\mathcal{H}}})WT_{i}^{*}h \\ &= K_{f,\varphi,T}^{(\mathcal{I})}T_{i}^{*}h \\ &= \bigl(B_{i}^{*}\otimes I_{\mathcal{D}_{C_{f,\varphi,T}}}\bigr)K_{f,\varphi ,T}^{(\mathcal{I})}h \\ &= \bigl(B_{i}^{*}\otimes I_{\mathcal{D}_{C_{f,\varphi,T}}}\bigr) (P_{\mathcal {N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{D}_{C_{f,\varphi ,T}}}|_{\widetilde{\mathcal{H}}})Wh\end{aligned} $$

for any \(h\in\mathcal{H}\) and \(i=1,\ldots,n\). Hence, we obtain that

$$ (P_{\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal {D}_{C_{f,\varphi,T}}}|_{\widetilde{\mathcal{H}}})\widetilde {T}_{i}^{*}z=\bigl(B_{i}^{*}\otimes I_{\mathcal{D}_{C_{f,\varphi,T}}} \bigr) (P_{\mathcal {N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{D}_{C_{f,\varphi ,T}}}|_{\widetilde{\mathcal{H}}})z $$
(3.15)

for any \(z\in\widetilde{\mathcal{H}}\) and \(i=1,\ldots,n\). Notice that \(P_{\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{D}_{C_{f,\varphi ,T}}}|_{\widetilde{\mathcal{H}}}\) is an injective operator. Then (3.15) uniquely determines each operator \(\widetilde{T}_{i}\), \(i=1,\ldots,n\).

(ii) First, assume that \(T=(T_{1},\ldots,T_{n})\in\mathcal{V}_{f,\varphi ,\mathcal{I}}^{\mathrm{pure}}(\mathcal{H})\). Due to Theorem 3.1, we know that the constrained Poisson kernel \(K_{f,\varphi,T}^{(\mathcal{I})}:\mathcal{H}\to\mathcal{N}_{f,\varphi ,\mathcal{I}}\otimes\mathcal{D}_{C_{f,\varphi,T}}\) is an isometry. Hence, \(K_{f,\varphi,T}^{(\mathcal{I})}(K_{f,\varphi,T}^{(\mathcal {I})})^{*}\) is the orthogonal projection from \(\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal {D}_{C_{f,\varphi,T}}\) onto \(K_{f,\varphi,T}^{(\mathcal{I})}\mathcal{H}\). According to Theorem 3.2, we deduce that \(\varTheta_{f,\varphi ,T}^{(\mathcal{I})}(\varTheta_{f,\varphi,T}^{(\mathcal{I})})^{*}\) is also a projection, which implies that \(\varTheta_{f,\varphi ,T}^{(\mathcal{I})}\) is a partial isometry. This shows that \(\varTheta _{f,\varphi,T}^{(\mathcal{I})}\) is an inner multi-analytic operator.

Conversely, if \(\varTheta_{f,\varphi,T}^{(\mathcal{I})}\) is an inner multi-analytic operator, then it is a partial isometry. Applying Theorem 3.2, we infer that \(K_{f,\varphi,T}^{(\mathcal{I})}\) is a partial isometry. Moreover, since T is in the noncommutative variety \(\mathcal{V}_{f,\varphi,\mathcal{I}}^{\mathrm{cnc}}(\mathcal{H})\), due to (3.5), we deduce that \(K_{f,\varphi,T}^{(\mathcal{I})}\) is an injective operator, which implies that \(K_{f,\varphi,T}^{(\mathcal{I})}\) is an isometry. Therefore, using Theorem 3.1, we deduce that T is in \(\mathcal{V}_{f,\varphi,\mathcal{I}}^{\mathrm{pure}}(\mathcal{H})\).

Now, we prove the last part of the theorem. Notice that \(u\oplus v\in \widetilde{\mathcal{K}}\) is in \(\widetilde{\mathcal{H}}\) if and only if

$$ \bigl\langle u\oplus v,\varTheta_{f,\varphi,T}^{(\mathcal{I})}x \oplus\Delta _{\varTheta_{C_{f,\varphi,T}}^{(\mathcal{I})}}x\bigr\rangle =0 $$
(3.16)

for any \(x\in\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal {D}_{C_{f,\varphi,T}^{*}}\). Note that condition (3.16) is equivalent to

$$ \bigl(\varTheta_{f,\varphi,T}^{(\mathcal{I})}\bigr)^{*}u+ \Delta_{\varTheta _{f,\varphi,T}^{(\mathcal{I})}}v=0. $$
(3.17)

Since the operator \(\Delta_{\varTheta_{f,\varphi,T}^{(\mathcal{I})}}\) is the orthogonal projection from \(\mathcal{N}_{f,\varphi,\mathcal {I}}\otimes\mathcal{D}_{C_{f,\varphi,T}^{*}}\) onto \([\operatorname{range} (\varTheta_{f,\varphi,T}^{(\mathcal {I})})^{*}]^{\perp}\), we have

$$\bigl(\varTheta_{f,\varphi,T}^{(\mathcal{I})}\bigr)^{*}u\perp \Delta_{\varTheta _{f,\varphi,T}^{(\mathcal{I})}}v. $$

Hence, (3.17) holds if and only if \((\varTheta_{f,\varphi,T}^{(\mathcal{I})})^{*}u=0\) and \(v=0\). Therefore, we conclude that

$$\widetilde{\mathcal{K}}=\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes \mathcal{D}_{C_{f,\varphi,T}} $$

and

$$\widetilde{\mathcal{H}}=(\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes \mathcal{D}_{C_{f,\varphi,T}})\ominus\varTheta_{f,\varphi,T}^{(\mathcal {I})}( \mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal {D}_{C_{f,\varphi,T}^{*}}). $$

According to (3.15), we infer that

$$\widetilde{T}_{i}=P_{\widetilde{\mathcal{H}}}(B_{i}\otimes I_{\mathcal {D}_{C_{f,\varphi,T}}})|_{\widetilde{\mathcal{H}}},\quad i=1,\ldots,n. $$

This completes the proof. □

Let \(\varPhi:\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal {H}_{1}\to\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{H}_{2}\) and \(\varPhi':\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{H}'_{1}\to \mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{H}'_{2}\) be two multi-analytic operators with respect to the constrained weighted shifts \(B_{1},\ldots,B_{n}\), i.e.,

$$\varPhi(B_{i}\otimes I_{\mathcal{H}_{1}})=(B_{i} \otimes I_{\mathcal {H}_{2}})\varPhi\quad\mbox{and}\quad \varPhi'(B_{i} \otimes I_{\mathcal{H}'_{1}})=(B_{i}\otimes I_{\mathcal {H}'_{2}}) \varPhi' $$

for any \(i=1,\ldots,n\). We say that Φ and \(\varPhi'\) coincide if there exist two unitary operators \(U_{j}\in B(\mathcal{H}_{j},\mathcal {H}_{j}')\), \(j=1,2\), such that

$$\varPhi'(I_{\mathcal{N}_{f,\varphi,\mathcal{I}}}\otimes U_{1})=(I_{\mathcal{N}_{f,\varphi,\mathcal{I}}} \otimes U_{2})\varPhi. $$

Applying Theorem 3.3, we can show that the constrained characteristic function \(\varTheta_{f,\varphi,T}^{(\mathcal{I})}\) is a complete unitary invariant for the n-tuples of operators in the noncommutative variety \(\mathcal{V}_{f,\varphi,\mathcal{I}}^{\mathrm{cnc}}(\mathcal{H})\).

Theorem 3.4

Let\(f:=\sum_{\alpha\in\mathbb{F}_{n}^{+}}a_{\alpha}Z_{\alpha}\)be a positive regular free holomorphic function, and let\(\varphi=(\varphi_{1},\ldots ,\varphi_{n})\)be ann-tuple of formal power series with model property. Let\(\mathcal{I} \neq H^{\infty}(\mathbb{D}_{f,\varphi})\)be a WOT-closed two-sided ideal of the noncommutative Hardy algebra\(H^{\infty}(\mathbb{D}_{f,\varphi})\). If\(T=(T_{1},\ldots,T_{n})\in\mathcal {V}_{f,\varphi,\mathcal{I}}^{\mathrm{cnc}}(\mathcal{H})\)and\(T'=(T'_{1},\ldots,T'_{n})\in\mathcal{V}_{f,\varphi,\mathcal {I}}^{\mathrm{cnc}}(\mathcal{H}')\), thenTand\(T'\)are unitarily equivalent if and only if their constrained characteristic functions\(\varTheta_{f,\varphi,T}^{(\mathcal{I})}\)and\(\varTheta _{f,\varphi,T'}^{(\mathcal{I})}\)coincide.

Proof

First, we assume that \(\varTheta_{f,\varphi,T}^{(\mathcal{I})}\) and \(\varTheta_{f,\varphi,T'}^{(\mathcal{I})}\) coincide. Then there are two unitary operators \(U_{1}:\mathcal{D}_{C_{f,\varphi,T}}\to\mathcal{D}_{C_{f,\varphi,T'}}\) and \(U_{2}:\mathcal{D}_{C_{f,\varphi,T}^{*}}\to\mathcal{D}_{C_{f,\varphi ,T'}^{*}}\) such that

$$(I_{\mathcal{N}_{f,\varphi,\mathcal{I}}}\otimes U_{1})\varTheta_{f,\varphi ,T}^{(\mathcal{I})}= \varTheta_{f,\varphi,T'}^{(\mathcal{I})}(I_{\mathcal {N}_{f,\varphi,\mathcal{I}}}\otimes U_{2}). $$

Consequently, we have

$$\Delta_{\varTheta_{f,\varphi,T}^{(\mathcal{I})}}=(I_{\mathcal {N}_{f,\varphi,\mathcal{I}}}\otimes U_{2})^{*} \Delta_{\varTheta_{f,\varphi ,T'}^{(\mathcal{I})}}(I_{\mathcal{N}_{f,\varphi,\mathcal{I}}}\otimes U_{2}) $$

and

$$(I_{\mathcal{N}_{f,\varphi,\mathcal{I}}}\otimes U_{2})\overline{\bigl[\Delta _{\varTheta_{f,\varphi,T}^{(\mathcal{I})}}(\mathcal{N}_{f,\varphi ,\mathcal{I}}\otimes\mathcal{D}_{C_{f,\varphi,T}^{*}}) \bigr]} =\overline{\bigl[\Delta_{\varTheta_{f,\varphi,T'}^{(\mathcal{I})}}(\mathcal {N}_{f,\varphi,\mathcal{I}} \otimes\mathcal{D}_{C_{f,\varphi,T'}^{*}})\bigr]}. $$

Now we define the unitary operator \(W:\widetilde{\mathcal{K}}\to \widetilde{\mathcal{K}}'\) by setting

$$W:=(I_{\mathcal{N}_{f,\varphi,\mathcal{I}}}\otimes U_{1})\oplus (I_{\mathcal{N}_{f,\varphi,\mathcal{I}}} \otimes U_{2}), $$

where \(\widetilde{\mathcal{K}}\) and \(\widetilde{\mathcal{K}}'\) were defined in Theorem 3.3. Notice that the operator \(\varPsi:\mathcal{N}_{f,\varphi,\mathcal {I}}\otimes\mathcal{D}_{C_{f,\varphi,T}^{*}}\to\widetilde{\mathcal{K}}\), defined by

$$\varPsi x:=\varTheta_{f,\varphi,T}^{(\mathcal{I})}x\oplus\Delta _{\varTheta_{f,\varphi,T}^{(\mathcal{I})}}x, \quad x\in\mathcal {N}_{f,\varphi,\mathcal{I}}\otimes \mathcal{D}_{C_{f,\varphi,T}^{*}}, $$

and the corresponding \(\varPsi':\mathcal{N}_{f,\varphi,\mathcal {I}}\otimes\mathcal{D}_{C_{f,\varphi,T'}^{*}}\to\widetilde{\mathcal {K}}'\) satisfy the following relations:

$$ W\varPsi(I_{\mathcal{N}_{f,\varphi,\mathcal{I}}}\otimes U_{2})^{*}= \varPsi' $$
(3.18)

and

$$ (I_{\mathcal{N}_{f,\varphi,\mathcal{I}}}\otimes U_{1})P_{\mathcal {N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{D}_{C_{f,\varphi ,T}}}^{\widetilde{\mathcal{K}}}W^{*} =P_{\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal {D}_{C_{f,\varphi,T'}}}^{\widetilde{\mathcal{K}}'}, $$
(3.19)

where \(P_{\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal {D}_{C_{f,\varphi,T}}}^{\widetilde{\mathcal{K}}}\) is the orthogonal projection from \(\widetilde{\mathcal{K}}\) onto \(\mathcal{N}_{f,\varphi,\mathcal {I}}\otimes\mathcal{D}_{C_{f,\varphi,T}}\). Hence, we have

$$\begin{aligned} W\widetilde{\mathcal{H}}&= W\widetilde{\mathcal{K}}\ominus W\varPsi ( \mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{D}_{C_{f,\varphi ,T}^{*}}) \\ &= \widetilde{\mathcal{K}}'\ominus\varPsi'(I_{\mathcal{N}_{f,\varphi ,\mathcal{I}}} \otimes U_{2}) (\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes \mathcal{D}_{C_{f,\varphi,T}^{*}}) \\ &= \widetilde{\mathcal{K}}'\ominus\varPsi'( \mathcal{N}_{f,\varphi ,\mathcal{I}}\otimes\mathcal{D}_{C_{f,\varphi,T'}^{*}}) \\ &= \widetilde{\mathcal{H}}', \end{aligned}$$

which implies that \(W|_{\widetilde{\mathcal{H}}}:\widetilde{\mathcal {H}}\to\widetilde{\mathcal{H}}'\) is unitary. On the other hand, for any \(i=1,\ldots,n\),

$$ \bigl(B_{i}^{*}\otimes I_{\mathcal{D}_{C_{f,\varphi,T'}}}\bigr) (I_{\mathcal {N}_{f,\varphi,\mathcal{I}}}\otimes U_{1})=(I_{\mathcal{N}_{f,\varphi ,\mathcal{I}}}\otimes U_{1}) \bigl(B_{i}^{*}\otimes I_{\mathcal{D}_{C_{f,\varphi,T}}}\bigr). $$
(3.20)

Now, we assume that \(\widetilde{T}:=(\widetilde{T}_{1},\ldots,\widetilde {T}_{n})\) and \(\widetilde{T}':=(\widetilde{T}'_{1},\ldots,\widetilde {T}'_{n})\) are the model operators provided by Theorem 3.3 for T and \(T'\), respectively. Therefore, applying (3.18), (3.19), and (3.20), we deduce that

$$\begin{aligned} P_{\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{D}_{C_{f,\varphi ,T'}}}^{\widetilde{\mathcal{K}}'}\widetilde{T}_{i}^{\prime *}Wz&= \bigl(B_{i}^{*}\otimes I_{\mathcal{D}_{C_{f,\varphi,T'}}}\bigr) P_{\mathcal{N}_{f,\varphi,\mathcal {I}}\otimes\mathcal{D}_{C_{f,\varphi,T'}}}^{\widetilde{\mathcal {K}}'}Wz \\ &= \bigl(B_{i}^{*}\otimes I_{\mathcal{D}_{C_{f,\varphi,T'}}}\bigr) (I_{\mathcal {N}_{f,\varphi,\mathcal{I}}}\otimes U_{1})P_{\mathcal{N}_{f,\varphi ,\mathcal{I}}\otimes\mathcal{D}_{C_{f,\varphi,T}}}^{\widetilde{\mathcal {K}}}z \\ &= (I_{\mathcal{N}_{f,\varphi,\mathcal{I}}}\otimes U_{1}) \bigl(B_{i}^{*} \otimes I_{\mathcal{D}_{C_{f,\varphi,T}}}\bigr)P_{\mathcal{N}_{f,\varphi,\mathcal {I}}\otimes\mathcal{D}_{C_{f,\varphi,T}}}^{\widetilde{\mathcal{K}}}z \\ &= (I_{\mathcal{N}_{f,\varphi,\mathcal{I}}}\otimes U_{1})P_{\mathcal {N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{D}_{C_{f,\varphi ,T}}}^{\widetilde{\mathcal{K}}} \widetilde{T}_{i}^{*}z \\ &= P_{\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal {D}_{C_{f,\varphi,T'}}}^{\widetilde{\mathcal{K}}'}W\widetilde{T}_{i}^{*}z \end{aligned}$$

for any \(z\in\widetilde{\mathcal{H}}\) and \(i=1,\ldots,n\). Using the fact that \(P_{\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal {D}_{C_{f,\varphi,T'}}}^{\widetilde{\mathcal{K}}'}\) is an injective operator, we infer that

$$(W|_{\widetilde{\mathcal{H}}})\widetilde{T}_{i}^{*}=\widetilde {T}_{i}^{\prime *}(W|_{\widetilde{\mathcal{H}}}),\quad i=1,\ldots,n. $$

Due to Theorem 3.3, it is obvious that T and \(T'\) are unitarily equivalent.

Conversely, let \(\varOmega:\mathcal{H}\to\mathcal{H}'\) be a unitary operator such that

$$T_{i}=\varOmega^{*}T_{i}'\varOmega\quad \mbox{for any } i=1,\ldots,n. $$

Note that \(T\in\mathcal{C}_{\varphi}^{\mathrm{SOT}}(\mathcal{H})\) or \(T\in\mathcal {C}_{\varphi}^{\mathrm{rad}}(\mathcal{H})\) and similar relations hold for \(T'\). Then we obtain

$$\varOmega\Delta_{C_{f,\varphi,T}}=\Delta_{C_{f,\varphi,T'}}\varOmega \quad \mbox{and}\quad \bigl( {\oplus}_{i=1}^{n}\varOmega\bigr) \Delta_{C_{f,\varphi ,T}^{*}}=\Delta_{C_{f,\varphi,T'}^{*}}\bigl( {\oplus}_{i=1}^{n} \varOmega\bigr). $$

Now we define the unitary operator by setting

$$U_{3}:=\varOmega|_{\mathcal{D}_{C_{f,\varphi,T}}}:\mathcal {D}_{C_{f,\varphi,T}} \to\mathcal{D}_{C_{f,\varphi,T'}} $$

and

$$U_{4}:=\bigl( {\oplus}_{i=1}^{n}\varOmega \bigr)|_{\mathcal{D}_{C_{f,\varphi ,T}^{*}}}:\mathcal{D}_{C_{f,\varphi,T}^{*}}\to\mathcal{D}_{C_{f,\varphi,T'}^{*}}. $$

A simple calculation shows that

$$(I_{\mathcal{N}_{f,\varphi,\mathcal{I}}}\otimes U_{3})\varTheta_{f,\varphi ,T}^{(\mathcal{I})}= \varTheta_{f,\varphi,T'}^{(\mathcal{I})}(I_{\mathcal {N}_{f,\varphi,\mathcal{I}}}\otimes U_{4}). $$

This completes the proof. □

Multivariable interpolation and invariant subspaces

In this section, we prove a Sarason-type commutant lifting theorem. As an application, we obtain the Nevanlinna–Pick-type interpolation result in our setting. Moreover, we provide a Beurling-type characterization of the joint invariant subspaces under the constrained weighted shifts \(B_{1},\ldots,B_{n}\).

For each \(i=1,\ldots,n\), we define the right multiplication operator \(R_{i}:\mathcal{F}_{f}^{2}\to\)\(\mathcal{F}_{f}^{2}\) by setting \(R_{i}\zeta=\zeta Z_{i}\), \(\zeta\in\mathcal{F}_{f}^{2}\). Using the results from [24], we know that \(R^{\infty}(\mathcal {D}_{f})\) is the WOT-closure of all polynomials in \(R_{1},\ldots,R_{n}\) and the identity. Moreover, we define the noncommutative Hardy algebra \(R^{\infty}(\mathbb {D}_{f,\varphi})\) to be the WOT-closure of all noncommutative polynomials in \(R_{Z_{1}},\ldots,R_{Z_{n}}\) and the identity.

The following result is a Sarason-type [32] commutant lifting theorem.

Theorem 4.1

Let\(f:=\sum_{\alpha\in\mathbb{F}_{n}^{+}}a_{\alpha}Z_{\alpha}\)be a positive regular free holomorphic function, and let\(\varphi=(\varphi_{1},\ldots ,\varphi_{n})\)be ann-tuple of formal power series with model property. Let\(\mathcal{I} \neq H^{\infty}(\mathbb{D}_{f,\varphi})\)be a WOT-closed two-sided ideal of the noncommutative Hardy algebra\(H^{\infty}(\mathbb{D}_{f,\varphi})\). For each\(j=1,2\), let\(\mathcal {K}_{j}\)be a Hilbert space, and let\(\mathcal{E}_{j}\subseteq\mathcal {N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{K}_{j}\)be an invariant subspace under\(B_{i}^{*}\otimes I_{\mathcal{K}_{j}}\), \(i=1,\ldots,n\). If\(X:\mathcal{E}_{1}\to\mathcal{E}_{2}\)is a bounded operator such that

$$X\bigl[P_{\mathcal{E}_{1}}(B_{i}\otimes I_{\mathcal{K}_{1}})|_{\mathcal {E}_{1}} \bigr]=\bigl[P_{\mathcal{E}_{2}}(B_{i}\otimes I_{\mathcal{K}_{2}})|_{\mathcal {E}_{2}} \bigr]X,\quad i=1,\ldots,n, $$

then there exists

$$\varPhi(C_{1},\ldots,C_{n})\in R^{\infty}( \mathcal{V}_{f,\varphi,\mathcal {I}})\mathrel{\overline{\otimes}}B(\mathcal{K}_{1}, \mathcal{K}_{2}) $$

such that

$$\varPhi(C_{1},\ldots,C_{n})^{*}\mathcal{E}_{2} \subseteq\mathcal{E}_{1},\qquad \varPhi (C_{1}, \ldots,C_{n})^{*}|_{\mathcal{E}_{2}}=X^{*}, \quad\textit{and}\quad \bigl\Vert \varPhi(C_{1},\ldots ,C_{n}) \bigr\Vert = \Vert X \Vert . $$

Proof

First, note that the subspace \(\mathcal{N}_{f,\varphi,\mathcal {I}}\otimes\mathcal{K}_{j}\) is invariant under \(M_{Z_{i}}^{*}\otimes I_{\mathcal{K}_{j}}\), and

$$\bigl(M_{Z_{i}}^{*}\otimes I_{\mathcal{K}_{j}}\bigr)|_{\mathcal{N}_{f,\varphi,\mathcal {I}}\otimes\mathcal{K}_{j}}=B_{i}^{*} \otimes I_{\mathcal{K}_{j}},\quad i=1,\ldots,n. $$

Since \(\mathcal{E}_{j}\subseteq\mathcal{N}_{f,\varphi,\mathcal {I}}\otimes\mathcal{K}_{j}\) is invariant under \(B_{1}^{*}\otimes I_{\mathcal {K}_{j}},\ldots,B_{n}^{*}\otimes I_{\mathcal{K}_{j}}\), it is also invariant under \(M_{Z_{1}}^{*}\otimes I_{\mathcal{K}_{j}},\ldots ,M_{Z_{n}}^{*}\otimes I_{\mathcal{K}_{j}}\), which implies that

$$\bigl(M_{Z_{i}}^{*}\otimes I_{\mathcal{K}_{j}}\bigr)|_{\mathcal{E}_{j}}= \bigl(B_{i}^{*}\otimes I_{\mathcal{K}_{j}}\bigr)|_{\mathcal{E}_{j}}, \quad i=1,\ldots,n. $$

Hence, we deduce that

$$X\bigl[P_{\mathcal{E}_{1}}(M_{Z_{i}}\otimes I_{\mathcal{K}_{1}})|_{\mathcal {E}_{1}} \bigr]=\bigl[P_{\mathcal{E}_{2}}(M_{Z_{i}}\otimes I_{\mathcal{K}_{2}})|_{\mathcal {E}_{2}} \bigr]X,\quad i=1,\ldots,n. $$

According to Theorem 5.1 of [28], there exists a bounded operator \(\varPhi:\mathbb{H}^{2}_{f}(\varphi)\otimes\mathcal{K}_{1}\to \mathbb{H}^{2}_{f}(\varphi)\otimes\mathcal{K}_{2}\) with the property

$$\varPhi(M_{Z_{i}}\otimes I_{\mathcal{K}_{1}})=(M_{Z_{i}} \otimes I_{\mathcal {K}_{2}})\varPhi,\quad i=1,\ldots,n, $$

and such that \(\varPhi^{*}\mathcal{E}_{2}\subseteq\mathcal{E}_{1}\), \(\varPhi ^{*}|_{\mathcal{E}_{2}}=X^{*}\), and \(\|\varPhi\|=\|X\|\). Since \(M_{\varphi_{i}}=\varphi_{i}(M_{Z_{1}},\ldots, M_{Z_{n}})\) for any \(i=1,\ldots,n\), we have

$$\varPhi(M_{\varphi_{i}}\otimes I_{\mathcal{K}_{1}})=(M_{\varphi_{i}} \otimes I_{\mathcal{K}_{2}})\varPhi,\quad i=1,\ldots,n. $$

Notice that

$$M_{\varphi_{i}}=U^{-1}V_{i}U, \quad i=1,\ldots,n. $$

Then we obtain

$$\varPhi\bigl(U^{-1}\otimes I_{\mathcal{K}_{1}}\bigr) (V_{i}\otimes I_{\mathcal {K}_{1}}) (U\otimes I_{\mathcal{K}_{1}})= \bigl(U^{-1}\otimes I_{\mathcal {K}_{2}}\bigr) (V_{i} \otimes I_{\mathcal{K}_{2}}) (U\otimes I_{\mathcal{K}_{2}})\varPhi $$

for any \(i=1,\ldots,n\). This shows that

$$\bigl[(U\otimes I_{\mathcal{K}_{2}})\varPhi\bigl(U^{-1}\otimes I_{\mathcal {K}_{1}}\bigr)\bigr](V_{i}\otimes I_{\mathcal{K}_{1}})=(V_{i} \otimes I_{\mathcal {K}_{2}})\bigl[(U\otimes I_{\mathcal{K}_{2}})\varPhi \bigl(U^{-1}\otimes I_{\mathcal{K}_{1}}\bigr)\bigr] $$

for any \(i=1,\ldots,n\). Due to the discussion of Proposition 1.11 from [24], we infer that

$$ \bigl[(U\otimes I_{\mathcal{K}_{2}})\varPhi \bigl(U^{-1}\otimes I_{\mathcal {K}_{1}}\bigr)\bigr]\in R^{\infty}(\mathcal{D}_{f})\mathrel{\overline{\otimes}}B(\mathcal {K}_{1},\mathcal{K}_{2}). $$
(4.1)

Using Proposition 4.2 in [28], we know

$$R^{\infty}(\mathbb{D}_{f,\varphi})=U^{-1}R^{\infty}( \mathcal{D}_{f})U. $$

Consequently, we infer that

$$\varPhi\in R^{\infty}(\mathbb{D}_{f,\varphi})\mathrel{\overline{\otimes}}B( \mathcal {K}_{1},\mathcal{K}_{2}). $$

Assume that \(\varPhi(R_{Z_{1}},\ldots,R_{Z_{n}}):=\varPhi\). This shows that we can find \(\varPhi(R_{Z_{1}},\ldots, R_{Z_{n}})\in R^{\infty}(\mathbb {D}_{f,\varphi})\mathrel{\overline{\otimes}}B(\mathcal{K}_{1},\mathcal{K}_{2})\) such that \(\varPhi(R_{Z_{1}},\ldots,R_{Z_{n}})^{*}\mathcal{E}_{2}\subseteq\mathcal{E}_{1}\),

$$ \varPhi(R_{Z_{1}},\ldots,R_{Z_{n}})^{*}|_{\mathcal{E}_{2}}=X^{*} \quad \mbox{and}\quad \bigl\Vert \varPhi(R_{Z_{1}}, \ldots,R_{Z_{n}}) \bigr\Vert = \Vert X \Vert . $$
(4.2)

Moreover, we assume that

$$\varPhi(C_{1},\ldots,C_{n}):=P_{\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes \mathcal{K}_{2}} \varPhi(R_{Z_{1}},\ldots,R_{Z_{n}})|_{\mathcal{N}_{f,\varphi ,\mathcal{I}}\otimes\mathcal{K}_{1}}. $$

Then we have \(\varPhi(C_{1},\ldots,C_{n})\in R^{\infty}(\mathcal{V}_{f,\varphi ,\mathcal{I}})\mathrel{\overline{\otimes}}B(\mathcal{K}_{1},\mathcal{K}_{2})\). Notice that

$$\varPhi(R_{Z_{1}},\ldots,R_{Z_{n}})^{*}(\mathcal{N}_{f,\varphi,\mathcal {I}} \otimes\mathcal{K}_{2})\subseteq\mathcal{N}_{f,\varphi,\mathcal {I}}\otimes \mathcal{K}_{1} $$

and \(\mathcal{E}_{j}\subseteq\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes \mathcal{K}_{j}\). Using (4.2), we obtain

$$\varPhi(C_{1},\ldots,C_{n})^{*}\mathcal{E}_{2} \subseteq\mathcal{E}_{1}\quad \mbox{and}\quad \varPhi(C_{1}, \ldots,C_{n})^{*}|_{\mathcal{E}_{2}}=X^{*}. $$

Applying again (4.2), we infer that

$$\Vert X \Vert \le \bigl\Vert \varPhi(C_{1}, \ldots,C_{n}) \bigr\Vert \le \bigl\Vert \varPhi(R_{Z_{1}}, \ldots ,R_{Z_{n}}) \bigr\Vert = \Vert X \Vert , $$

which shows that

$$\bigl\Vert \varPhi(C_{1},\ldots,C_{n}) \bigr\Vert = \Vert X \Vert . $$

This completes the proof. □

Applying Theorem 4.1, we can obtain the following Nevanlinna–Pick-type interpolation result in our setting.

Theorem 4.2

Let\(f:=\sum_{\alpha\in\mathbb{F}_{n}^{+}}a_{\alpha}Z_{\alpha}\)be a positive regular free holomorphic function, and let\(\varphi=(\varphi_{1},\ldots ,\varphi_{n})\)be ann-tuple of formal power series with model property. Let\(\mathcal{I} \neq H^{\infty}(\mathbb{D}_{f,\varphi})\)be a WOT-closed two-sided ideal of the noncommutative Hardy algebra\(H^{\infty}(\mathbb{D}_{f,\varphi})\). Let\(\lambda_{1},\ldots,\lambda_{k}\)bekdistinct points in\(\mathcal{V}_{f,\varphi,\mathcal{I}}^{<}(\mathbb {C})\), and let\(A_{1},\ldots,A_{k}\in B(\mathcal{K})\). Then there exists\(\varPhi(C_{1},\ldots,C_{n})\in R^{\infty}(\mathcal{V}_{f,\varphi,\mathcal {I}})\mathrel{\overline{\otimes}}B(\mathcal{K})\)such that

$$\bigl\Vert \varPhi(C_{1},\ldots,C_{n}) \bigr\Vert \le1\quad \textit{and}\quad \varPhi(\lambda _{j})=A_{j}, \quad j=1,\ldots,k, $$

if and only if the operator matrix

$$ \bigl[K_{f,\varphi}(\lambda_{i}, \lambda_{j}) \bigl(I_{\mathcal{K}}-A_{i}A_{j}^{*} \bigr) \bigr]_{k\times k} $$
(4.3)

is positive semidefinite, where

$$K_{f,\varphi}(\lambda_{i},\lambda_{j}):= \frac{\sqrt{1-\sum_{ \vert \alpha \vert \ge 1}a_{\alpha} \vert \varphi_{\alpha}(\lambda_{i}) \vert ^{2}}\sqrt{1-\sum_{ \vert \alpha \vert \ge 1}a_{\alpha} \vert \varphi_{\alpha}(\lambda_{j}) \vert ^{2}}}{ 1-\sum_{ \vert \alpha \vert \ge1}a_{\alpha}[\varphi(\lambda_{i})]_{\alpha}[\overline {\varphi(\lambda_{j})}]_{\alpha}}. $$

Proof

Let \(\lambda_{j}:=(\lambda_{j_{1}},\ldots,\lambda_{j_{n}})\), \(j=1,\ldots,k\), be k distinct points in \(\mathcal{V}_{f,\varphi,\mathcal{I}}^{<}(\mathbb {C})\), and let

$$ Z_{f,\varphi}^{(\lambda_{j})}:=\sqrt{1-\sum _{ \vert \alpha \vert \ge1}a_{\alpha}\bigl\vert \varphi _{\alpha}(\lambda_{j}) \bigr\vert ^{2}} \biggl(\sum_{\alpha\in\mathbb{F}_{n}^{+}}b_{\alpha}\bigl[\overline {\varphi(\lambda_{j})}\bigr]_{\alpha}\varphi_{\alpha}\biggr),\quad j=1,\ldots,k, $$
(4.4)

where the coefficients \(b_{\alpha}\), \(\alpha\in\mathbb{F}_{n}^{+}\), are given by relation (2.1). Since φ has model property, we have

$$M_{\varphi_{i}}=\varphi_{i}(M_{Z_{1}}, \ldots,M_{Z_{n}}),\quad i=1,\ldots,n, $$

where \((M_{Z_{1}},\ldots,M_{Z_{n}})\) is either in the set \(\mathcal {C}_{\varphi}^{\mathrm{SOT}}(\mathbb{H}_{f}^{2}(\varphi))\) or \(\mathcal{C}_{\varphi}^{\mathrm{rad}}(\mathbb{H}_{f}^{2}(\varphi))\). Due to Proposition 4.2 of [28], for any \(\omega\in\mathcal {I}\subseteq H^{\infty}(\mathbb{D}_{f,\varphi})\), there exists \(\chi=\sum_{\alpha\in\mathbb{F}_{n}^{+}}c_{\alpha}V_{\alpha}\in F^{\infty}(\mathcal {D}_{f})\) such that

$$ \omega=\text{SOT-}\lim_{r\to1}\sum _{k=0}^{\infty}\sum _{|\alpha |=k}c_{\alpha}r^{|\alpha|}M_{\varphi_{\alpha}}. $$
(4.5)

Using (4.4) and (4.5), we infer that

$$\bigl\langle Z_{f,\varphi}^{(\lambda_{j})},\omega(1)\bigr\rangle _{f,\varphi}=0 \quad\mbox{for any } \omega\in\mathcal{I} \mbox{ and } j=1,\ldots,k. $$

Since \(\mathcal{I}\) is a WOT-closed two-sided ideal of \(H^{\infty}(\mathbb {D}_{f,\varphi})\), we obtain

$$\mathcal{M}_{f,\varphi,\mathcal{I}}=\overline{\mathcal{I}(1)}. $$

This shows that

$$Z_{f,\varphi}^{(\lambda_{j})}\in\mathcal{N}_{f,\varphi,\mathcal {I}},\quad j=1, \ldots,k. $$

According to Theorem 4.4 of [28], we have

$$M_{Z_{i}}^{*}Z_{f,\varphi}^{(\lambda_{j})}=\overline{ \lambda_{ji}}Z_{f,\varphi }^{(\lambda_{j})},\quad i=1,\ldots,n; j=1,\ldots,k. $$

Moreover, notice that

$$B_{i}^{*}|_{ \mathcal{N}_{f,\varphi,\mathcal{I}}}=M_{Z_{i}}^{*}|_{ \mathcal {N}_{f,\varphi,\mathcal{I}}}, \quad i=1,\ldots,n. $$

Hence, we deduce that the subspace

$$\mathcal{M}:=\operatorname{span}\bigl\{ Z_{f,\varphi}^{(\lambda_{j})}: j=1, \ldots,k\bigr\} $$

is invariant under \(B_{i}^{*}\) for any \(i=1,\ldots,n\), and \(\mathcal {M}\subseteq\mathcal{N}_{f,\varphi,\mathcal{I}}\). Now, we define the operators \(X_{i}\in B(\mathcal{M}\otimes\mathcal{K})\) by setting

$$X_{i}:=P_{\mathcal{M}}B_{i}|_{\mathcal{M}} \otimes I_{\mathcal{K}}, \quad i=1,\ldots,n. $$

Note that \(Z_{f,\varphi}^{(\lambda_{1})},\ldots,Z_{f,\varphi}^{(\lambda _{k})}\) are linearly independent. Then we can define an operator \(T\in B(\mathcal{M}\otimes\mathcal{K})\) by setting

$$T^{*}\bigl(Z_{f,\varphi}^{(\lambda_{j})}\otimes h\bigr)=Z_{f,\varphi}^{(\lambda _{j})} \otimes A_{j}^{*}h $$

for any \(h\in\mathcal{K}\) and \(j=1,\ldots,k\). A simple calculation shows that

$$TX_{i}=X_{i}T,\quad i=1,\ldots,n. $$

Taking into account that \(\mathcal{M}\otimes\mathcal{K}\) is a co-invariant subspace under \(B_{i}\otimes I_{\mathcal{K}}\), \(i=1,\ldots,n\). Due to Theorem 4.1, we can find \(\varPhi(R_{Z_{1}},\ldots ,R_{Z_{n}})\in R^{\infty}(\mathbb{D}_{f,\varphi})\mathrel{\overline{\otimes}}B(\mathcal{K})\) such that

$$\varPhi(C_{1},\ldots,C_{n}):=P_{\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes \mathcal{K}} \varPhi(R_{Z_{1}},\ldots,R_{Z_{n}})|_{\mathcal{N}_{f,\varphi ,\mathcal{I}}\otimes\mathcal{K}}\in R^{\infty}(\mathcal{V}_{f,\varphi ,\mathcal{I}})\mathrel{\overline{\otimes}}B(\mathcal{K}) $$

has the properties

$$\varPhi(C_{1},\ldots,C_{n})^{*}(\mathcal{M}\otimes \mathcal{K})\subseteq \mathcal{M}\otimes\mathcal{K},\qquad \varPhi(C_{1}, \ldots ,C_{n})^{*}|_{\mathcal{M}\otimes\mathcal{K}}=T^{*}, $$

and

$$\bigl\Vert \varPhi(C_{1},\ldots,C_{n}) \bigr\Vert = \Vert T \Vert . $$

In what follows, we prove

$$R_{Z_{i}}^{*}Z_{f,\varphi}^{(\lambda)}=\overline{ \lambda_{i}}Z_{f,\varphi }^{(\lambda)}\quad\mbox{for any } \lambda\in\mathbb{D}_{f,\varphi }^{< }(\mathbb{C}) \mbox{ and } i=1,\ldots, n, $$

where \(Z_{f,\varphi}^{(\lambda)}\) is given by relation (4.4). Indeed, a straightforward computation reveals that

$$R_{\varphi_{\beta}}^{*}\varphi_{\alpha}=\left \{ \textstyle\begin{array}{l@{\quad}l} \frac{b_{\gamma}}{b_{\alpha}}\varphi_{\gamma}, & \alpha=\gamma\widetilde {\beta}, \\ 0, & \text{otherwise}. \end{array}\displaystyle \right . $$

Consequently, we obtain

$$\begin{aligned} R_{\varphi_{i}}^{*}Z_{f,\varphi}^{(\lambda)}&= R_{\varphi_{i}}^{*} \sqrt{1-\sum_{ \vert \alpha \vert \ge1}a_{\alpha}\bigl\vert \varphi_{\alpha}(\lambda) \bigr\vert ^{2}}\biggl(\sum _{\alpha\in \mathbb{F}_{n}^{+}}b_{\alpha}\bigl[\overline{\varphi( \lambda)}\bigr]_{\alpha}\varphi _{\alpha}\biggr) \\ &= \sqrt{1-\sum_{ \vert \alpha \vert \ge1}a_{\alpha}\bigl\vert \varphi_{\alpha}(\lambda) \bigr\vert ^{2}}\biggl( \sum_{\gamma\in\mathbb{F}_{n}^{+}}\frac{b_{\gamma}}{b_{\gamma g_{i}}}b_{\gamma g_{i}} \overline{\bigl[\varphi(\lambda)\bigr]}_{\gamma g_{i}}\varphi_{\gamma}\biggr) \\ &= \sqrt{1-\sum_{ \vert \alpha \vert \ge1}a_{\alpha}\bigl\vert \varphi_{\alpha}(\lambda) \bigr\vert ^{2}}\biggl( \sum_{\gamma\in\mathbb{F}_{n}^{+}}b_{\gamma}\overline{\bigl[ \varphi(\lambda)\bigr]}_{\gamma g_{i}}\varphi_{\gamma}\biggr) \\ &= \overline{\varphi_{i}(\lambda)} \biggl[\sqrt{1-\sum _{ \vert \alpha \vert \ge 1}a_{\alpha}\bigl\vert \varphi_{\alpha}( \lambda) \bigr\vert ^{2}}\biggl(\sum_{\gamma\in\mathbb {F}_{n}^{+}}b_{\gamma}\overline{\bigl[\varphi(\lambda)\bigr]}_{\gamma}\varphi_{\gamma}\biggr) \biggr] \\ &= \overline{\varphi_{i}(\lambda)}Z_{f,\varphi}^{(\lambda)} \end{aligned}$$

for any \(i=1,\ldots,n\). Moreover, due to the proof of Theorem 2.1 from [28], we have

$$R_{Z_{i}}=\psi_{i}(R_{\varphi_{1}}, \ldots,R_{\varphi_{n}})=\operatorname {SOT} {\text{-}}\lim_{r\to1}\psi_{i}(rR_{\varphi_{1}},\ldots,rR_{\varphi_{n}}) $$

for any \(i=1,\ldots,n\). Hence, we conclude that

$$\psi_{i}(R_{\varphi_{1}},\ldots,R_{\varphi_{n}})^{*}Z_{f,\varphi}^{(\lambda )}=\overline{\psi_{i}(\varphi(\lambda))}Z_{f,\varphi}^{(\lambda)} $$

for any \(i=1,\ldots,n\). Since \(\lambda\in\mathbb{D}_{f,\varphi }^{<}(\mathbb{C})\), we obtain \(\lambda_{i}=\psi_{i}(\varphi(\lambda))\) for any \(i=1, \ldots,n\). Therefore, we infer that

$$R_{Z_{i}}^{*}Z_{f,\varphi}^{(\lambda)}=\psi_{i}(R_{\varphi_{1}}, \ldots ,R_{\varphi_{n}})^{*}Z_{f,\varphi}^{(\lambda)}=\overline{ \psi_{i}\bigl(\varphi (\lambda)\bigr)}Z_{f,\varphi}^{(\lambda)}= \overline{\lambda_{i}}Z_{f,\varphi }^{(\lambda)} $$

for any \(i=1,\ldots,n\). This proves our assertion. Since \(\lambda _{1},\ldots,\lambda_{k}\) are k distinct points in \(\mathcal{V}_{f,\varphi ,\mathcal{I}}^{<}(\mathbb{C})\subseteq\mathbb{D}_{f,\varphi}^{<}(\mathbb{C})\), we have \(R_{Z_{i}}^{*}Z_{f,\varphi}^{(\lambda_{j})}=\overline{\lambda _{ji}}Z_{f,\varphi}^{(\lambda_{j})}\), \(i=1,\ldots,n\); \(j=1, \ldots,k\). This shows that

$$\nu(R_{Z_{1}},\ldots,R_{Z_{n}})^{*}Z_{f,\varphi}^{(\lambda_{j})}= \overline{\nu (\lambda_{j})}Z_{f,\varphi}^{(\lambda_{j})} $$

for any \(\nu(R_{Z_{1}},\ldots,R_{Z_{n}})\in R^{\infty}(\mathbb{D}_{f,\varphi })\). Hence, we deduce that

$$ \varPhi(R_{Z_{1}},\ldots,R_{Z_{n}})^{*} \bigl(Z_{f,\varphi}^{(\lambda_{j})}\otimes h\bigr)=Z_{f,\varphi}^{(\lambda_{j})} \otimes\varPhi(\lambda_{j})^{*}h,\quad j=1,\ldots,k. $$
(4.6)

Using (4.6), we obtain

$$\begin{aligned} & \bigl\langle \varPhi(C_{1},\ldots,C_{n})^{*} \bigl(Z_{f,\varphi}^{(\lambda _{j})}\otimes x\bigr),Z_{f,\varphi}^{(\lambda_{j})} \otimes y\bigr\rangle \\ &\quad= \bigl\langle \varPhi(R_{Z_{1}},\ldots,R_{Z_{n}})^{*} \bigl(Z_{f,\varphi}^{(\lambda _{j})}\otimes x\bigr),Z_{f,\varphi}^{(\lambda_{j})} \otimes y\bigr\rangle \\ &\quad= \bigl\langle Z_{f,\varphi}^{(\lambda_{j})}\otimes\varPhi(\lambda _{j})^{*}x,Z_{f,\varphi}^{(\lambda_{j})}\otimes y\bigr\rangle \\ &\quad= \bigl\langle Z_{f,\varphi}^{(\lambda_{j})},Z_{f,\varphi}^{(\lambda _{j})} \bigr\rangle _{f,\varphi}\bigl\langle \varPhi(\lambda_{j})^{*}x,y \bigr\rangle \end{aligned}$$
(4.7)

for any \(x,y\in\mathcal{K}\) and \(j=1,\ldots,k\). Moreover, notice that

$$ \bigl\langle T^{*}\bigl(Z_{f,\varphi}^{(\lambda_{j})} \otimes x\bigr),Z_{f,\varphi }^{(\lambda_{j})}\otimes y\bigr\rangle =\bigl\langle Z_{f,\varphi}^{(\lambda _{j})},Z_{f,\varphi}^{(\lambda_{j})} \bigr\rangle _{f,\varphi}\bigl\langle A_{j}^{*}x,y\bigr\rangle $$
(4.8)

for any \(x,y\in\mathcal{K}\) and \(j=1,\ldots,k\). Since \(\varphi(\lambda _{1}),\ldots,\varphi(\lambda_{k})\) are in the strict noncommutative domain \(\mathcal{D}_{f,<}(\mathbb{C})\), we infer that

$$ \bigl\langle Z_{f,\varphi}^{(\lambda_{i})},Z_{f,\varphi}^{(\lambda_{j})} \bigr\rangle _{f,\varphi}=\frac{\sqrt{1-\sum_{ \vert \alpha \vert \ge1}a_{\alpha} \vert \varphi_{\alpha}(\lambda_{j}) \vert ^{2}}\sqrt{1-\sum_{ \vert \alpha \vert \ge1}a_{\alpha} \vert \varphi_{\alpha}(\lambda_{i}) \vert ^{2}}}{ 1-\sum_{ \vert \alpha \vert \ge1}a_{\alpha}[\varphi(\lambda_{j})]_{\alpha}[\overline {\varphi(\lambda_{i})}]_{\alpha}}\neq0 $$
(4.9)

for any \(i,j=1,\ldots,k\). Hence, applying (4.7), (4.8), and (4.9), we conclude that \(\varPhi(\lambda_{j})=A_{j}\), \(j=1,\ldots,k\), if and only if \(\varPhi(C_{1},\ldots,C_{n})^{*}|_{\mathcal{M}\otimes\mathcal {K}}=T^{*}\).

Since \(\|\varPhi(C_{1},\ldots,C_{n})\|=\|T\|\), it is clear that

$$\bigl\Vert \varPhi(C_{1},\ldots,C_{n}) \bigr\Vert \le1\quad\mbox{if and only if}\quad TT^{*}\le I_{\mathcal{M}\otimes\mathcal{K}}. $$

On the other hand, for any \(h_{1},\ldots,h_{k}\in\mathcal{K}\), we have

$$\begin{aligned} & \Biggl\langle \sum_{j=1}^{k} Z_{f,\varphi}^{(\lambda_{j})}\otimes h_{j},\sum _{j=1}^{k} Z_{f,\varphi}^{(\lambda_{j})} \otimes h_{j}\Biggr\rangle -\Biggl\langle T^{*}\Biggl(\sum _{j=1}^{k} Z_{f,\varphi}^{(\lambda_{j})} \otimes h_{j}\Biggr),T^{*}\Biggl(\sum_{j=1}^{k} Z_{f,\varphi}^{(\lambda_{j})}\otimes h_{j}\Biggr)\Biggr\rangle \\ &\quad= \sum_{i,j=1}^{k}\bigl\langle Z_{f,\varphi}^{(\lambda_{i})},Z_{f,\varphi }^{(\lambda_{j})}\bigr\rangle _{f,\varphi} \bigl\langle \bigl(I_{\mathcal{K}}-A_{j}A_{i}^{*}\bigr)h_{i},h_{j} \bigr\rangle \\ &\quad= \sum_{i,j=1}^{k} K_{f,\varphi}( \lambda_{j},\lambda_{i}) \bigl\langle \bigl(I_{\mathcal{K}}-A_{j}A_{i}^{*} \bigr)h_{i},h_{j}\bigr\rangle . \end{aligned}$$

Consequently, we deduce that \(\|\varPhi(C_{1},\ldots,C_{n})\|\le1\) if and only if matrix (4.3) is positive semidefinite. This completes the proof. □

The following result is a noncommutative multivariable version of a result of Rosenblum and Rovnyak [31].

Theorem 4.3

Let\(f:=\sum_{\alpha\in\mathbb{F}_{n}^{+}}a_{\alpha}Z_{\alpha}\)be a positive regular free holomorphic function, and let\(\varphi=(\varphi_{1},\ldots ,\varphi_{n})\)be ann-tuple of formal power series with model property. Let\(\mathcal{I} \neq H^{\infty}(\mathbb{D}_{f,\varphi})\)be a WOT-closed two-sided ideal of the noncommutative Hardy algebra\(H^{\infty}(\mathbb{D}_{f,\varphi})\). If\(X\in B(\mathcal{N}_{f,\varphi ,\mathcal{I}}\otimes\mathcal{K})\)is a self-adjoint operator, then the following statements are equivalent:

  1. (i)

    \(\varPhi_{f,\varphi,B\otimes I_{\mathcal{K}}}(X)\le X\), where\(B\otimes I_{\mathcal{K}}:=(B_{1}\otimes I_{\mathcal{K}},\ldots,B_{n}\otimes I_{\mathcal{K}})\);

  2. (ii)

    there are a Hilbert space\(\mathcal{G}\)and a multi-analytic operator\(\varPhi:\mathcal{N}_{f,\varphi,\mathcal {I}}\otimes\mathcal{G}\to\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes \mathcal{K}\)with respect to the constrained weighted shifts\(B_{1},\ldots,B_{n}\)such that\(X=\varPhi\varPhi^{*}\).

Proof

First, we prove that (i) (ii). Since \((B_{1},\ldots,B_{n})\) is a pure n-tuple of operators in the noncommutative variety \(\mathcal{V}_{f,\varphi,\mathcal{I}}(\mathcal {N}_{f,\varphi,\mathcal{I}})\) and

$$- \Vert X \Vert \varPhi_{f,\varphi,B\otimes I_{\mathcal{K}}}^{m}(I)\le\varPhi _{f,\varphi,B\otimes I_{\mathcal{K}}}^{m}(X)\le \Vert X \Vert \varPhi_{f,\varphi ,B\otimes I_{\mathcal{K}}}^{m}(I), $$

we deduce that

$$\text{SOT-}\lim_{m\to\infty}\varPhi_{f,\varphi,B\otimes I_{\mathcal{K}}}^{m}(X)=0. $$

Notice that

$$\varPhi_{f,\varphi,B\otimes I_{\mathcal{K}}}^{m}(X)\le\varPhi_{f,\varphi ,B\otimes I_{\mathcal{K}}}^{m-1}(X) \le\cdots\le X,\quad m\in\mathbb{N}. $$

Then we obtain \(X\ge0\). Let \(\mathcal{M}:=\overline{\operatorname{range} X^{\frac{1}{2}}}\) and define

$$ Q_{i}\bigl(X^{\frac{1}{2}}\xi \bigr):=X^{\frac{1}{2}}\bigl(\varphi_{i}(B)^{*}\otimes I_{\mathcal{K}}\bigr)\xi,\quad\xi\in\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes \mathcal{K}, $$
(4.10)

for any \(i=1,\ldots,n\). Note that

$$\begin{aligned} \sum_{ \vert \alpha \vert \ge1}a_{\alpha}\bigl\Vert Q_{\widetilde{\alpha}}\bigl(X^{\frac{1}{2}}\xi\bigr) \bigr\Vert ^{2}&\le \sum_{ \vert \alpha \vert \ge1} \bigl\Vert \sqrt{a_{\alpha}}X^{\frac{1}{2}}\bigl(\bigl[\varphi (B) \bigr]_{\alpha}^{*}\otimes I_{\mathcal{K}}\bigr)\xi \bigr\Vert ^{2} \\ &= \bigl\langle \varPhi_{f,\varphi,B\otimes I_{\mathcal{K}}}(X)\xi,\xi\bigr\rangle \\ &\le \bigl\Vert X^{\frac{1}{2}}\xi \bigr\Vert ^{2}\end{aligned} $$

for any \(\xi\in\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{K}\). Hence, we obtain that

$$a_{g_{i}} \bigl\Vert Q_{i}X^{\frac{1}{2}}\xi \bigr\Vert ^{2}\le \bigl\Vert X^{\frac{1}{2}}\xi \bigr\Vert ^{2},\quad\xi\in \mathcal{N}_{f,\varphi,\mathcal{I}}\otimes \mathcal{K}, $$

for any \(i=1,\ldots,n\). Since f is a positive regular free holomorphic function, each operator \(Q_{i}\), \(i=1,\ldots,n\), can be uniquely extended to a bounded operator (also denoted by \(Q_{i}\)) on \(\mathcal{M}\). Denoting \(A_{i}:=Q_{i}^{*}\) for any \(i=1,\ldots,n\), we have

$$\sum_{|\alpha|\ge1}a_{\alpha}A_{\alpha}A_{\alpha}^{*}\le I_{\mathcal{M}}, $$

where the convergence is in the weak operator topology. Setting \(\phi _{A}(X):=\sum_{|\alpha|\ge1}a_{\alpha}A_{\alpha}X A_{\alpha}^{*}\) (the convergence is in the weak operator topology) and using (4.10), we infer that

$$\begin{aligned} \bigl\langle \phi_{A}^{m}(I)X^{\frac{1}{2}} \xi,X^{\frac{1}{2}}\xi\bigr\rangle ={} & \bigl\langle \varPhi_{f,\varphi,B\otimes I_{\mathcal{K}}}^{m}(X) \xi,\xi\bigr\rangle \\ \le{}& \Vert X \Vert \bigl\langle \varPhi_{f,\varphi,B\otimes I_{\mathcal{K}}}^{m}(I) \xi ,\xi\bigr\rangle \end{aligned}$$

for any \(\xi\in\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{K}\), which implies that

$$\text{SOT-}\lim_{m\to\infty}\phi_{A}^{m}(I)=0. $$

This shows that \(A:=(A_{1},\ldots,A_{n})\) is a pure n-tuple of operators in \(\mathcal{D}_{f}(\mathcal{M})\). According to Proposition 4.2 of [28], we know that \(\mathcal{I}\) is a WOT-closed two-sided ideal of \(H^{\infty}(\mathbb{D}_{f,\varphi})\) if and only if there is a WOT-closed two-sided ideal J of \(F^{\infty}(\mathcal{D}_{f})\) such that

$$\mathcal{I}=\bigl\{ \chi\bigl(\varphi(M_{Z})\bigr):\chi\in J\bigr\} . $$

Taking into account that

$$ X^{\frac{1}{2}}A_{i}=\bigl( \varphi_{i}(B)\otimes I_{\mathcal{K}}\bigr)X^{\frac{1}{2}},\quad i=1,\ldots,n. $$
(4.11)

Then, for any \(\chi\in J\), we obtain

$$X^{\frac{1}{2}}\chi(rA_{1},\ldots,rA_{n})= \bigl(\chi\bigl(r\varphi_{1}(B),\ldots,r\varphi _{n}(B) \bigr)\otimes I_{\mathcal{K}}\bigr)X^{\frac{1}{2}} $$

for any \(r\in(0,1)\). Moreover, since \((A_{1},\ldots,A_{n})\) is a pure n-tuple of operators in the noncommutative domain \(\mathcal{D}_{f}(\mathcal{M})\) and \((\varphi _{1}(B),\ldots,\varphi_{n}(B))\) is also a pure n-tuple of operators in \(\mathcal{D}_{f}(\mathcal{N}_{f,\varphi,\mathcal{I}})\), using \(F^{\infty}(\mathcal{D}_{f})\)-functional calculus (see [24]), we have

$$X^{\frac{1}{2}}\chi(A_{1},\ldots,A_{n})= \bigl(\chi\bigl(\varphi_{1}(B),\ldots,\varphi _{n}(B) \bigr)\otimes I_{\mathcal{K}}\bigr)X^{\frac{1}{2}}=0 $$

for any \(\chi\in J\). Since \(X^{\frac{1}{2}}\) is an injective operator on \(\mathcal{M}\), we infer that

$$\chi(A_{1},\ldots,A_{n})=0\quad\mbox{for any } \chi\in J. $$

Consequently, we deduce that \((A_{1},\ldots,A_{n})\) is a pure n-tuple of operators in the noncommutative variety \(\mathcal{V}_{f,J}(\mathcal {M})\), where

$$\mathcal{V}_{f,J}(\mathcal{M}):=\bigl\{ (T_{1}, \ldots,T_{n})\in\mathcal {D}_{f}(\mathcal{M}): \chi(T_{1},\ldots,T_{n})=0 \mbox{ for any } \chi\in J\bigr\} . $$

Applying the appropriate result from [24], we know that the noncommutative Poisson kernel \(K_{f,A}:\mathcal{M}\to\mathbb {H}_{f}^{2}(\varphi)\otimes\mathcal{G}\) (\(\mathcal{G}\) is an appropriate Hilbert space) defined by

$$K_{f,A}h:=\sum_{\alpha\in\mathbb{F}_{n}^{+}}b_{\alpha}\varphi_{\alpha}\otimes \Delta_{f,A}A_{\alpha}^{*}h, \quad h\in\mathcal{M}, $$

where \(\Delta_{f,A}:=(I-\sum_{|\alpha|\ge1}a_{\alpha}A_{\alpha}A_{\alpha}^{*})^{\frac{1}{2}}\) is an isometry with the properties that

$$K_{f,A}(\mathcal{M})\subseteq N_{f,\varphi,\mathcal{I}}\otimes\mathcal {G} \quad\mbox{and}\quad K_{f,A}^{*}(M_{\varphi_{i}}\otimes I_{\mathcal{G}})=A_{i}K_{f,A}^{*} $$

for any \(i=1,\ldots,n\). Now we define

$$\varPhi:=X^{\frac{1}{2}} K_{f,A,\mathcal{I}}^{*}:\mathcal{N}_{f,\varphi,\mathcal{I}} \otimes\mathcal {G}\to\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{K}, $$

where the constrained Poisson kernel \(K_{f,A,\mathcal{I}}:\mathcal{M}\to \mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{G}\) is defined by

$$K_{f,A,\mathcal{I}}:=(P_{\mathcal{N}_{f,\varphi,\mathcal{I}}}\otimes I_{\mathcal{G}}) K_{f,A}. $$

Since φ has the model property, we have

$$M_{\varphi_{i}}=\varphi_{i}(M_{Z_{1}}, \ldots,M_{Z_{n}}),\quad i=1,\ldots,n, $$

where \((M_{Z_{1}},\ldots,M_{Z_{n}})\) is either in the set \(\mathcal {C}_{\varphi}^{\mathrm{SOT}}(\mathbb{H}_{f}^{2}(\varphi))\) or \(\mathcal{C}_{\varphi}^{\mathrm{rad}}(\mathbb{H}_{f}^{2}(\varphi))\). Hence, we obtain

$$ K_{f,A,\mathcal{I}}^{*}\bigl(\varphi_{i}(B)\otimes I_{\mathcal{G}}\bigr)=A_{i}K_{f,A,\mathcal{I}}^{*},\quad i=1, \ldots,n. $$
(4.12)

Therefore, using (4.11) and (4.12), we infer that

$$\begin{aligned} \varPhi\bigl(\varphi_{i}(B)\otimes I_{\mathcal{G}}\bigr)&= X^{\frac {1}{2}}K_{f,A,\mathcal{I}}^{*}\bigl(\varphi_{i}(B) \otimes I_{\mathcal{G}}\bigr)=X^{\frac {1}{2}}A_{i}K_{f,A,\mathcal{I}}^{*} \\ &= \bigl(\varphi_{i}(B)\otimes I_{\mathcal{K}} \bigr)X^{\frac{1}{2}}K_{f,A,\mathcal {I}}^{*}=\bigl(\varphi_{i}(B) \otimes I_{\mathcal{K}}\bigr)\varPhi \end{aligned}$$

for any \(i=1,\ldots,n\). On the other hand, notice that

$$\begin{aligned} B_{i}&= P_{\mathcal{N}_{f,\varphi,\mathcal{I}}}M_{Z_{i}}|_{\mathcal {N}_{f,\varphi,\mathcal{I}}} \\ &= P_{\mathcal{N}_{f,\varphi,\mathcal{I}}}\psi_{i}\bigl(\varphi_{1}(M_{Z}), \ldots ,\varphi_{n}(M_{Z})\bigr)|_{\mathcal{N}_{f,\varphi,\mathcal{I}}} \\ &= \psi_{i}\bigl(\varphi_{1}(B),\ldots, \varphi_{n}(B)\bigr) \end{aligned}$$

for any \(i=1,\ldots,n\). Then we conclude that each operator \(B_{i}\), \(i=1,\ldots,n\), is in the SOT-closure of all polynomials in \(\varphi _{1}(B),\ldots, \varphi_{n}(B)\) and the identity. Consequently, we obtain that

$$\varPhi(B_{i}\otimes I_{\mathcal{G}})=(B_{i} \otimes I_{\mathcal{K}})\varPhi ,\quad i=1,\ldots,n. $$

This shows that Φ is a multi-analytic operator with respect to the constrained weighted shifts \(B_{1},\ldots,B_{n}\). Moreover, since the constrained Poisson kernel \(K_{f,A,\mathcal{I}}\) is an isometry, we deduce that

$$\varPhi\varPhi^{*}=X^{\frac{1}{2}}K_{f,A,\mathcal{I}}^{*}K_{f,A,\mathcal {I}}X^{\frac{1}{2}}=X. $$

Now, we prove that (ii) (i). Note that \((B_{1},\ldots ,B_{n})\in\mathcal{V}_{f,\varphi,\mathcal{I}}(\mathcal{N}_{f,\varphi ,\mathcal{I}})\). Then we have

$$\begin{aligned} \varPhi_{f,\varphi,B\otimes I_{\mathcal{K}}}(X)&= \sum_{|\alpha|\ge 1}a_{\alpha}\bigl(\bigl[\varphi(B)\bigr]_{\alpha}\otimes I_{\mathcal{K}} \bigr)X\bigl(\bigl[\varphi (B)\bigr]_{\alpha}\otimes I_{\mathcal{K}} \bigr)^{*} \\ &= \sum_{|\alpha|\ge1}a_{\alpha}\bigl(\bigl[ \varphi(B)\bigr]_{\alpha}\otimes I_{\mathcal{K}}\bigr)\varPhi \varPhi^{*}\bigl(\bigl[\varphi(B)\bigr]_{\alpha}\otimes I_{\mathcal{K}}\bigr)^{*} \\ &= \varPhi\biggl(\sum_{|\alpha|\ge1}a_{\alpha}\bigl(\bigl[\varphi(B)\bigr]_{\alpha}\otimes I_{\mathcal{G}}\bigr) \bigl(\bigl[\varphi(B)\bigr]_{\alpha}\otimes I_{\mathcal{G}} \bigr)^{*}\biggr)\varPhi^{*} \\ &\le \varPhi\varPhi^{*}=X, \end{aligned}$$

where the convergence is in the weak operator topology. This completes the proof. □

As an application, we obtain a Beurling-type characterization of the invariant subspaces under the constrained weighted shifts \(B_{1},\ldots,B_{n}\).

Theorem 4.4

Let\(f:=\sum_{\alpha\in\mathbb{F}_{n}^{+}}a_{\alpha}Z_{\alpha}\)be a positive regular free holomorphic function, and let\(\varphi=(\varphi_{1},\ldots ,\varphi_{n})\)be ann-tuple of formal power series with model property. Let\(\mathcal{I} \neq H^{\infty}(\mathbb{D}_{f,\varphi})\)be a WOT-closed two-sided ideal of the noncommutative Hardy algebra\(H^{\infty}(\mathbb{D}_{f,\varphi})\). A subspace\(\mathcal{M}\subseteq \mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{K}\)is invariant under\(B_{i}\otimes I_{\mathcal{K}}\), \(i=1,\ldots,n\), if and only if there are a Hilbert space\(\mathcal{G}\)and an inner multi-analytic operator

$$\varPhi:\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{G}\to\mathcal {N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{K} $$

with respect to the constrained weighted shifts\(B_{1},\ldots,B_{n}\)such that

$$\mathcal{M}=\varPhi[\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{G}]. $$

Proof

First, we assume that \(\mathcal{M}\subseteq\mathcal{N}_{f,\varphi ,\mathcal{I}}\otimes\mathcal{K}\) is invariant under \(B_{1}\otimes I_{\mathcal{K}},\ldots,B_{n}\otimes I_{\mathcal{K}}\). Notice that

$$P_{\mathcal{M}}(B_{i}\otimes I_{\mathcal{K}})P_{\mathcal{M}}=(B_{i} \otimes I_{\mathcal{K}})P_{\mathcal{M}},\quad i=1,\ldots,n, $$

and \((B_{1},\ldots,B_{n})\in\mathcal{V}_{f,\varphi,\mathcal{I}}(\mathcal {N}_{f,\varphi,\mathcal{I}})\). Then we have

$$\begin{aligned} \varPhi_{f,\varphi,B\otimes I_{\mathcal{K}}}(P_{\mathcal{M}})={}& P_{\mathcal{M}} \biggl(\sum_{|\alpha|\ge1}a_{\alpha}\bigl(\bigl[ \varphi(B)\bigr]_{\alpha}\otimes I_{\mathcal{K}} \bigr)P_{\mathcal{M}}\bigl(\bigl[\varphi(B)\bigr]_{\alpha}^{*} \otimes I_{\mathcal{K}}\bigr)\biggr)P_{\mathcal{M}} \\ \le{}& P_{\mathcal{M}}\biggl(\sum_{|\alpha|\ge1}a_{\alpha}\bigl(\bigl[\varphi(B)\bigr]_{\alpha}\otimes I_{\mathcal{K}} \bigr) \bigl(\bigl[\varphi(B)\bigr]_{\alpha}^{*}\otimes I_{\mathcal{K}}\bigr)\biggr)P_{\mathcal{M}} \\ ={}& P_{\mathcal{M}}\biggl(\sum_{|\alpha|\ge1}a_{\alpha}\bigl[\varphi(B)\bigr]_{\alpha}\bigl[\varphi (B)\bigr]_{\alpha}^{*} \otimes I_{\mathcal{K}}\biggr)P_{\mathcal{M}} \\ \le{}& P_{\mathcal{M}}.\end{aligned} $$

According to Theorem 4.3, there are a Hilbert space \(\mathcal{G}\) and a multi-analytic operator

$$\varPhi:\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{G}\to \mathcal{N}_{f,\varphi,\mathcal{I}} \otimes\mathcal{K} $$

with respect to the constrained weighted shifts \(B_{1},\ldots,B_{n}\) such that \(P_{\mathcal{M}}=\varPhi\varPhi^{*}\). Moreover, since \(P_{\mathcal{M}}\) is an orthogonal projection, we deduce that Φ is a partial isometry and \(\mathcal{M}=\varPhi[\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{G}]\). The converse is obvious. This completes the proof. □

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Acknowledgements

The authors wish to thank the anonymous referees and the editor for their useful comments.

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This work is supported by the National Natural Science Foundation of China (No. 11771340).

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Hu, J., Wang, M. & Wang, W. Constrained characteristic functions, multivariable interpolation, and invariant subspaces. J Inequal Appl 2020, 146 (2020). https://doi.org/10.1186/s13660-020-02412-x

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MSC

  • 46L52
  • 47A45
  • 46L07
  • 46T25

Keywords

  • Noncommutative Poisson transform
  • Characteristic function
  • Universal model
  • Interpolation