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# Fredholmness of multiplication of a weighted composition operator with its adjoint on \(H^{2}\) and \(A_{\alpha}^{2}\)

*Journal of Inequalities and Applications*
**volume 2018**, Article number: 23 (2018)

## Abstract

In this paper, we obtain that \(C_{\psi,\varphi}^{\ast}\) is bounded below on \(H^{2}\) or \(A_{\alpha}^{2}\) if and only if \(C_{\psi,\varphi }\) is invertible. Moreover, we investigate the Fredholm operators \(C_{\psi_{1},\varphi_{1}}C_{\psi_{2},\varphi_{2}}^{\ast}\) and \(C_{\psi_{1},\varphi_{1}}^{\ast}C_{\psi_{2},\varphi_{2}}\) on \(H^{2}\) and \(A_{\alpha}^{2}\).

## 1 Introduction

Let \(\mathbb{D}\) denote the open unit disk in the complex plane. The Hardy space, denoted \(H^{2}(\mathbb{D})=H^{2}\), is the set of all analytic functions *f* on \(\mathbb{D}\) satisfying the norm condition

The space \(H^{\infty}(\mathbb{D})=H^{\infty}\) consists of all analytic and bounded functions on \(\mathbb{D}\) with supremum norm \(\| f\|_{\infty}=\sup_{z \in\mathbb{D}}|f(z)|\).

For \(\alpha> -1\), the weighted Bergman space \(A^{2}_{\alpha}(\mathbb {D})=A^{2}_{\alpha}\) is the set of functions *f* analytic in \(\mathbb {D}\) with

where *dA* is the normalized area measure in \(\mathbb{D}\). The case where \(\alpha=0\) is known as the (unweighted) Bergman space, often simply denoted by \(A^{2}\).

Let *φ* be an analytic map from the open unit disk \(\mathbb{D}\) into itself. The operator that takes the analytic map *f* to \(f \circ \varphi\) is a composition operator and is denoted by \(C_{\varphi}\). A natural generalization of a composition operator is an operator that takes *f* to \(\psi\cdot f \circ\varphi\), where *ψ* is a fixed analytic map on \(\mathbb{D}\). This operator is aptly named a weighted composition operator and is usually denoted by \(C_{\psi,\varphi}\). More precisely, if *z* is in the unit disk, then \((C_{\psi,\varphi}f)(z)=\psi(z)f(\varphi(z))\). For some results on weighted composition and related operators on the weighted Bergman and Hardy spaces, see, for example, [1–14].

If *ψ* is a bounded analytic function on the open unit disk, then the multiplication operator \(M_{\psi}\) defined by \(M_{\psi }(f)(z)=\psi(z)f(z)\) is a bounded operator on \(H^{2}\) and \(A_{\alpha }^{2}\) and \(\|M_{\psi}(f)\|_{\gamma} \leq\|\psi\|_{\infty}\|f\| _{\gamma}\) when \(\gamma=1\) for \(H^{2}\) and \(\gamma=\alpha+2\) for \(A_{\alpha}^{2}\). Let *P* denote the orthogonal projection of \(L^{2}(\partial\mathbb{D})\) onto \(H^{2}\). For each \(b \in{L^{\infty}(\partial\mathbb{D})}\), the Toeplitz operator \(T_{b}\) acts on \(H^{2}\) by \(T_{b}(f)=P(bf)\). Also suppose that \(P_{\alpha}\) is the orthogonal projection of \(L^{2}({\mathbb{D}}, dA_{\alpha})\) onto \(A^{2}_{\alpha}\). For each function \(w \in L^{\infty}(\mathbb{D})\), the Toeplitz operator \(T_{w}\) on \(A^{2}_{\alpha}\) is defined by \(T_{w}(f)=P_{\alpha}(wf)\). Since *P* and \(P_{\alpha}\) are bounded on \(H^{2}\) and \(A^{2}_{\alpha }\), respectively, the Toeplitz operators are bounded.

Let \(w \in\mathbb{D}\), and let *H* be a Hilbert space of analytic functions on \(\mathbb{D}\). Let \(e_{w}\) be the point evaluation at *w*, that is, \(e_{w}(f)=f(w)\) for \(f \in H\). If \(e_{w}\) is a bounded linear functional on *H*, then the Riesz representation theorem implies that there is a function (usually denoted \(K_{w}\)) in *H* that induces this linear functional, that is, \(e_{w}(f)=\langle f,K_{w}\rangle\). In this case, the functions \(K_{w}\) are called the reproducing kernels, and the functional Hilbert space is also called a reproducing kernel Hilbert space. Both the weighted Bergman spaces and the Hardy space are reproducing kernel Hilbert spaces, where the reproducing kernel for evaluation at *w* is given by \(K_{w}(z)=(1-\overline{w}z)^{-\gamma}\) for \(z,w \in\mathbb{D}\), with \(\gamma=1\) for \(H^{2}\) and \(\gamma=\alpha+2\) for \(A_{\alpha }^{2}\). Let \(k_{w}\) denote the normalized reproducing kernel given by \(k_{w}(z)=K_{w}(z)/\|K_{w}\|_{\gamma}\), where \(\|K_{w}\|_{\gamma}^{2}=(1-|w|^{2})^{-\gamma}\).

Suppose that *H* and \(H'\) are Hilbert spaces and \(A:H \rightarrow H'\) is a bounded operator. The operator *A* is said to be left semi-Fredholm if there are a bounded operator \(B:H'\rightarrow H\) and a compact operator *K* on *H* such that \(BA=I+K\). Analogously, *A* is right semi-Fredholm if there are a bounded operator \(B':H'\rightarrow H\) and a compact operator \(K'\) on \(H'\) such that \(AB'=I+K'\). An operator *A* is said to be Fredholm if it is both left and right semi-Fredholm. It is not hard to see that *A* is left semi-Fredholm if and only if \(A^{\ast}\) is right semi-Fredholm. Hence *A* is Fredholm if and only if \(A^{\ast}\) is Fredholm. Note that an invertible operator is Fredholm. By using the definition of a Fredholm operator it is not hard to see that if the operators *A* and *B* are Fredholm on a Hilbert space *H*, then *AB* is also Fredholm on *H*. The Fredholm composition operators on \(H^{2}\) were first identified by Cima et al. [15] and later by a different and more general method by Bourdon [2]. Cima et al. [15] proved that only the invertible composition operators on \(H^{2}\) are Fredholm. Moreover, MacCluer [16] characterized Fredholm composition operators on a variety of Hilbert spaces of analytic functions in both one and several variables. Recently, Fredholm composition operators on various spaces of analytic functions have been studied (see [13] and [14]).

The automorphisms of \(\mathbb{D}\), that is, the one-to-one analytic maps of the disk onto itself, are just the functions \(\varphi(z)=\lambda\frac{a-z}{1-\overline{a}z}\) with \(|\lambda|=1\) and \(|a| < 1\). We denote the class of automorphisms of \(\mathbb{D}\) by \(\operatorname{Aut}(\mathbb{D})\). Automorphisms of \(\mathbb{D}\) take \(\partial\mathbb{D}\) onto \(\partial\mathbb {D}\). It is known that \(C_{\varphi}\) is Fredholm on the Hardy space if and only if \(\varphi\in\operatorname{Aut}(\mathbb{D})\) (see [2]).

An analytic map *φ* of the disk to itself is said to have a finite angular derivative at a point *ζ* on the boundary of the disk if there exists a point *η*, also on the boundary of the disk, such that the nontangential limit as \(z \rightarrow\zeta\) of the difference quotient \((\eta-\varphi(z))/(\zeta-z)\) exists as a finite complex value. We write \(\varphi'(\zeta)=\angle\lim_{z \rightarrow\zeta} \frac{\eta -\varphi(z)}{\zeta-z}\).

In the second section, we investigate Fredholm and invertible weighted composition operators. In Theorem 2.7, we show that the operator \(C_{\psi,\varphi}^{\ast}\) is bounded below on \(H^{2}\) or \(A^{2}_{\alpha}\) if and only if \(C_{\psi,\varphi}\) is invertible.

In the third section, we investigate the Fredholm operators \(C_{\psi _{1},\varphi_{1}}C_{\psi_{2},\varphi_{2}}^{\ast}\) and \(C_{\psi _{1},\varphi_{1}}^{\ast}C_{\psi_{2},\varphi_{2}}\) on \(H^{2}\) and \(A^{2}_{\alpha}\).

## 2 Bounded below operators \(C_{\psi,\varphi}^{\ast}\)

Let *H* be a Hilbert space. The set of all bounded operators from *H* into itself is denoted by \(B(H)\). We say that an operator \(A \in B(H)\) is bounded below if there is a constant \(c > 0\) such that \(c\| h \| \leq \|A(h)\|\) for all \(h \in H\).

If *f* is defined on a set *V* and if there is a positive constant *m* such that \(|f(z)| \geq m\) for all *z* in *V*, then we say that *f* is bounded away from zero on *V*. In particular, we say that *ψ* is bounded away from zero near the unit circle if there are \(\delta> 0\) and \(\epsilon> 0\) such that

Suppose that *T* belongs to \(B(H)\). We denote the spectrum of *T*, the essential spectrum of *T*, the approximate point spectrum of *T*, and the point spectrum of *T* by \(\sigma(T)\), \(\sigma_{e}(T)\), \(\sigma _{\mathrm{ap}}(T)\) and \(\sigma_{p}(T)\), respectively. Moreover, the left essential spectrum of *T* is denoted by \(\sigma_{\mathrm{le}}(T)\).

Suppose that *φ* is an analytic self-map of \(\mathbb{D}\). For almost all \(\zeta \in\partial\mathbb{D}\), we define \(\varphi(\zeta )=\lim_{r \rightarrow1}\varphi(r\zeta)\) (the statement of the existence of this limit can be found in [17, Theorem 2.2]). If *f* is a bounded analytic function on the unit disk such that \(|f(e^{i\theta})|=1\) almost everywhere, then we call *f* an inner function. We know that if *φ* is inner, then \(C_{\varphi}\) is bounded below on \(H^{2}\), and therefore \(C_{\varphi}\) has a closed range (see [17, Theorem 3.8]).

Now we state the following simple and well-known lemma, and we frequently use it in this paper.

### Lemma 2.1

*Let*
\(C_{\psi,\varphi}\)
*be a bounded operator on*
\(H^{2}\)
*or*
\(A^{2}_{\alpha}\). *Then*
\(C_{\psi,\varphi}^{\ast}K_{w}=\overline{\psi(w)}K_{\varphi(w)}\)
*for all*
\(w \in\mathbb{D}\).

In this paper, for convenience, we assume that \(\gamma=1\) for \(H^{2}\) and \(\gamma=\alpha+2\) for \(A_{\alpha}^{2}\).

### Lemma 2.2

*Suppose that*
*A*
*and*
*B*
*are two bounded operators on a Hilbert space*
*H*. *If*
*AB*
*is a Fredholm operator*, *then*
*B*
*is left semi*-*Fredholm*.

### Proof

Suppose that *AB* is a Fredholm operator. Then there are a bounded operator *C* and a compact operator *K* such that \(CAB=I+K\). Therefore *B* is left semi-Fredholm. □

Zhao [13] characterized Fredholm weighted composition operators on \(H^{2}\). Also, Zhao [14] found necessary conditions of *φ* and *ψ* for a weighted composition operator \(C_{\psi ,\varphi}\) on \(A_{\alpha}^{2}\) to be Fredholm. In the following proposition, we obtain a necessary and sufficient condition for \(C_{\psi,\varphi}\) to be Fredholm on \(H^{2}\) and \(A_{\alpha}^{2}\). Then we use it to find when \(C_{\psi_{1},\varphi_{1}}^{\ast}C_{\psi _{2},\varphi_{2}}\) and \(C_{\psi_{1},\varphi_{1}}C_{\psi_{2},\varphi _{2}}^{\ast}\) are Fredholm. The idea of the proof of the next proposition is different from [13] and [14].

### Proposition 2.3

*The operator*
\(C^{\ast}_{\psi,\varphi}\)
*is left semi*-*Fredholm on*
\(H^{2}\)
*or*
\(A_{\alpha}^{2}\)
*if and only if*
\(\varphi\in\operatorname{Aut}(\mathbb{D})\)
*and*
\(\psi\in H^{\infty}\)
*is bounded away from zero near the unit circle*. *Under these conditions*, \(C_{\psi,\varphi}\)
*is a Fredholm operator*.

### Proof

Let \(C_{\psi,\varphi}\) be Fredholm on \(H^{2}\) or \(A_{\alpha}^{2}\). Assume that *ψ* is not bounded away from zero near the unit circle. Then for each positive integer *n*, there is \(x_{n} \in\mathbb{D}\) such that \(1-1/n < |x_{n}| < 1\) and \(|\psi (x_{n})| < 1/n\). Then there exist a subsequence \(\{x_{n_{m}}\}\) and \(\zeta\in\partial\mathbb{D}\) such that \(x_{n_{m}} \rightarrow\zeta \) as \(m \rightarrow\infty\). Since \(\psi(x_{n_{m}})\rightarrow0\) as \(m\rightarrow\infty\), by Lemma 2.1 we see that

where the last equality follows from the fact that \(\liminf\frac {1-|\varphi(x_{n_{k}})|}{1-|x_{n_{k}}|} \neq0\) (see [17, Corollary 2.40]). Since \(k_{x_{n_{m}}}\) tends to zero weakly as \(m \rightarrow\infty\) (see [17, Theorem 2.17]), by [18, Theorem 2.3, p. 350], \(C_{\psi,\varphi}^{\ast}\) is not left semi-Fredholm. This is a contradiction. Hence *ψ* is bounded away from zero near the unit circle. Denote the inner product on \(H^{2}\) or \(A_{\alpha}^{2}\) by by \(\langle\cdot,\cdot\rangle_{\gamma}\), where \(\gamma=1\) for \(H^{2}\) and \(\gamma=\alpha+2\) for \(A_{\alpha }^{2}\). Define the bounded linear functional \(F_{\psi}\) by \(F_{\psi }(f)=\langle f,\psi\rangle_{\gamma}\) for each *f* that belongs to \(H^{2}\) or \(A_{\alpha}^{2}\), where \(\gamma=1\) for \(H^{2}\) and \(\gamma =\alpha+2\) for \(A_{\alpha}^{2}\). We know that, for each \(\zeta\in \partial\mathbb{D}\), \(K_{r\zeta}/\|K_{r\zeta}\|_{\gamma}\) tends to zero weakly as \(r \rightarrow1\). Then

and so \(|\psi(r\zeta)|/\|K_{r\zeta}\|_{\gamma} \rightarrow0\) as \(r \rightarrow1\). Now, we show that *φ* is inner. For each \(\zeta\in\partial \mathbb{D}\) such that \(\varphi(\zeta):=\lim_{r\rightarrow1}\varphi (r\zeta)\) exists, by Lemma 2.1 we have

Since \(|\psi(r\zeta)|/\|K_{r\zeta}\|_{\gamma} \rightarrow0\) as \(r \rightarrow1\), if \(\varphi(\zeta) \notin\partial\mathbb{D}\), then \(\lim_{r \rightarrow1}\|C_{\psi,\varphi}^{\ast}k_{r\zeta}\| _{\gamma}=0\), which is a contradiction (see [18, Theorem 2.3, p. 350]). Hence *φ* is inner. Since \(C_{\psi,\varphi}^{\ast }\) is left semi-Fredholm, by Lemma 2.2, \(T_{\psi}^{\ast}\) is left semi-Fredholm. Then \(\operatorname{dim} \operatorname{ker} T_{\psi}^{\ast}\) is finite. It follows from Lemma 2.1 that *ψ* has only finite zeroes in \(\mathbb {D}\). If *φ* is constant on \(\mathbb{D}\), then \(\operatorname{dim} \operatorname{ker} C_{\psi,\varphi}^{\ast}= \operatorname{dim} (\operatorname{ran} C_{\psi,\varphi})^{\bot}=\infty\), a contradiction. If \(\varphi(a)=\varphi(b)\) for some \(a,b \in\mathbb{D}\) with \(a \neq b\), then by using the idea similar to that used in [2, Lemma] there exist infinite sets \(\{a_{n}\}\) and \(\{b_{n}\}\) in \(\mathbb{D}\) which are disjoint and such that \(\varphi(a_{n})= \varphi(b_{n})\). We can assume that \(\psi(a_{n})\psi(b_{n}) \neq0\) because *ψ* has only finite zeroes in \(\mathbb{D}\). By Lemma 2.1 we see that

Therefore, \(K_{a_{n}}/\overline{\psi(a_{n})}-K_{b_{n}}/\overline {\psi(b_{n})} \in\operatorname{ker} C_{\psi,\varphi}^{\ast}\). It is not hard to see that \(\{K_{a_{n}}/\overline{\psi (a_{n})}-K_{b_{n}}/\overline{\psi(b_{n})}\}\) is a linearly independent set in the kernel of \(C_{\psi,\varphi}^{\ast}\), and so we have our desired contradiction. Hence *φ* must be univalent. Then [17, Corollary 3.28] implies that *φ* is an automorphism of \(\mathbb{D}\). Since \(C_{\psi,\varphi}C_{\varphi }^{-1}=M_{\psi}\) is a bounded multiplication operator on \(H^{2}\) and \(A^{2}_{\alpha}\), by [19, p. 215], \(\psi\in H^{\infty}\).

Conversely, suppose that \(\varphi\in\operatorname{Aut}(\mathbb{D})\) and \(\psi\in H^{\infty}\) is bounded away from zero near the unit circle. Since \(C_{\varphi}\) is invertible, \(C_{\varphi}\) has a closed range. Since \(\psi\not\equiv0\), \(\operatorname{ker} T_{\psi}=(0)\). We infer that \(T_{\psi}\) has a closed range by [18, Corollary 2.4, p. 352], [20, Theorem 3], and [12, Theorem 8], so by [18, Proposition 6.4, p. 208], \(T_{\psi}\) is bounded below. We claim that \(C_{\psi,\varphi}\) has a closed range. This can be seen as follows. Suppose that \(\{h_{n}\}\) is a sequence such that \(\{C_{\psi ,\varphi}(h_{n})\}\) converges to *f* as \(n\rightarrow\infty\). Since \(T_{\psi}\) has a closed range, \(\{C_{\psi,\varphi}(h_{n})\}\) converges to \(T_{\psi}g\) for some *g* as \(n\rightarrow\infty\). Since \(T_{\psi}\) is bounded below, there is a constant \(c > 0\) such that \(\|T_{\psi}(C_{\varphi}(h_{n})-g)\| \geq c\|C_{\varphi }(h_{n})-g\|\). Therefore \(C_{\varphi}(h_{n}) \rightarrow g\) as \(n \rightarrow\infty\). There exists *h* such that \(C_{\varphi}(h)=g\) because \(C_{\varphi}\) has a closed range. Hence \(f=C_{\psi,\varphi }(h)\), as desired. Hence \(\operatorname{ran} C_{\psi,\varphi}\) is closed and \(\operatorname{ker} C_{\psi,\varphi}=(0)\). [20, Theorem 3] and [12, Theorem 10] imply that \(T_{\psi}\) is Fredholm, and so \(\operatorname{ker} T_{\psi}^{\ast }\) is finite dimensional. Since \(\varphi\in \operatorname{Aut}(\mathbb{D})\), it is not hard to see that

Therefore, \(\operatorname{dim} \operatorname{ker} C_{\psi,\varphi}^{\ast} < \infty\), and the conclusion follows from [18, Corollary 2.4, p. 352]. □

In the next proposition, we give a necessary condition of *ψ* for an operator \(C_{\psi,\varphi}^{\ast}\) to be bounded below on \(H^{2}\) and \(A^{2}_{\alpha}\). Then we use Proposition 2.4 to obtain all invertible weighted composition operators on \(H^{2}\) and \(A^{2}_{\alpha}\).

### Proposition 2.4

*Let*
*ψ*
*be an analytic map of*
\(\mathbb {D}\), *and let*
*φ*
*be an analytic self*-*map of*
\(\mathbb{D}\). *If*
\(C_{\psi,\varphi}^{\ast}\)
*is bounded below on*
\(H^{2}\)
*or*
\(A^{2}_{\alpha}\), *then*
\(\psi\in H^{\infty}\)
*is bounded away from zero on*
\(\mathbb{D}\), *and*
\(\varphi\in\operatorname{Aut}(\mathbb{D})\).

### Proof

Let \(\varphi\equiv d\) for some \(d \in\mathbb{D}\). Since \(C_{\psi,\varphi}^{\ast}\) is bounded below, there is a constant \(c > 0\) such that \(\|C_{\psi,\varphi}^{\ast} f\|_{\gamma} \geq c\|f\| _{\gamma}\) for all *f*. Then for each \(w \in\mathbb{D}\), by Lemma 2.1, \(\|C_{\psi,\varphi}^{\ast}K_{w}\|_{\gamma}=|\psi(w)|\|K_{d}\| _{\gamma} \geq c\|K_{w}\|_{\gamma}\). Therefore, for each \(w \in\mathbb{D}\),

It is easy to see that *ψ* is bounded away from zero on \(\mathbb {D}\). Now assume that *φ* is not a constant function. Suppose that *ψ* is not bounded away from zero on \(\mathbb{D}\). Therefore, there exist a sequence \(\{x_{n}\}\) in \(\mathbb{D}\) and \(a \in\overline{\mathbb{D}}\) such that \(x_{n} \rightarrow a\) and \(|\psi (x_{n})| \rightarrow0\) as \(n \rightarrow\infty\). First, suppose that \(a \in\mathbb{D}\). By Lemma 2.1 we have

Since \(C_{\psi,\varphi}^{\ast}\) is bounded below, \(0 \geq c\|k_{a}\| _{\gamma}=c\), a contradiction. Now assume that \(a \in\partial\mathbb {D}\). It is not hard to see that there is a subsequence \(\{x_{n_{m}}\}\) such that \(\{\varphi(x_{n_{m}})\}\) converges. By Lemma 2.1 we see that

If \(\{\varphi(x_{n_{m}})\}\) converges to a point in \(\mathbb{D}\), then (1) is equal to zero. Now assume that \(\{\varphi(x_{n_{m}})\}\) converges to a point in \(\partial\mathbb{D}\). If *φ* has a finite angular derivative at *a*, then by the Julia-Carathéodory theorem we have

which shows that (1) is equal to zero. If *φ* does not have a finite angular derivative at *a*, then

so again (1) is equal to zero. Since \(C_{\psi,\varphi}^{\ast}\) is bounded below and \(\|k_{x_{n_{m}}}\|_{\gamma}=1\), we have \(c=0\), is a contradiction. Therefore, *ψ* is bounded away from zero on \(\mathbb {D}\). Since by [18, Proposition 6.4, p. 208], \(0 \notin\sigma _{\mathrm{ap}}(C_{\psi,\varphi}^{\ast})\), we have that \(\lim_{r \rightarrow 1}\|C_{\psi,\varphi}^{\ast}k_{r\zeta}\|_{\gamma} \neq0\) for all \(\zeta\in\partial\mathbb{D}\). We employ the idea of the proof of Proposition 2.3 to see that *φ* is a univalent inner function. Thus \(\varphi\in\operatorname{Aut}(\mathbb{D})\) (see [17, Corollary 3.28]). Moreover, since \(C_{\psi,\varphi}\) is a bounded operator, as we saw in the proof of Proposition 2.3, we conclude that \(\psi\in H^{\infty}\), and the proposition follows. □

Bourdon [21, Theorem 3.4] obtained the following corollary; we give another proof (see also [22, Theorem 2.0.1]).

### Corollary 2.5

*Let*
*ψ*
*be an analytic map of*
\(\mathbb {D}\), *and let*
*φ*
*be an analytic self*-*map of*
\(\mathbb{D}\). *The weighted composition operator*
\(C_{\psi,\varphi}\)
*is invertible on*
\(H^{2}\)
*or*
\(A_{\alpha}^{2}\)
*if and only if*
\(\varphi\in\operatorname{Aut}(\mathbb{D})\)
*and*
\(\psi\in H^{\infty}\)
*is bounded away from zero on*
\(\mathbb{D}\).

### Proof

Let \(C_{\psi,\varphi}\) be invertible. Then \(C_{\psi ,\varphi}^{\ast}\) is bounded below. The conclusion follows from Proposition 2.4. The reverse direction is trivial since \(C_{\varphi}\) and \(T_{\psi}\) are invertible. □

Note that if \(C_{\psi,\varphi}\) is invertible, then \(C_{\psi,\varphi }^{\ast}\) is bounded below. Hence by Proposition 2.4 and Corollary 2.5 we can see that \(C_{\psi,\varphi}^{\ast}\) is bounded below if and only if \(C_{\psi,\varphi}\) is invertible.

The algebra \(A(\mathbb{D})\) consists of all continuous functions on the closure of \(\mathbb{D}\) that are analytic on \({\mathbb{D}}\). In the next corollary, we find some Fredholm weighted composition operators that are not invertible.

### Corollary 2.6

*Suppose that*
\(\varphi\in\operatorname{Aut}(\mathbb {D})\)
*and*
\(\psi\in A(\mathbb{D})\). *Assume that*
\(\{z \in\mathbb{D}: \psi(z)=0\}\)
*is a nonempty finite set and*
\(\psi(z) \neq0\)
*for all*
\(z \in\partial \mathbb{D}\). *Then*
\(C_{\psi,\varphi}\)
*is Fredholm*, *but it is not invertible*.

### Proof

It is easy to see that *ψ* is bounded away from zero near the unit circle. Therefore the result follows from Proposition 2.3 and Corollary 2.5. □

### Theorem 2.7

*Suppose that*
*ψ*
*is an analytic map of*
\(\mathbb{D}\)
*and*
*φ*
*is an analytic self*-*map of*
\(\mathbb{D}\). *The operator*
\(C_{\psi,\varphi}^{\ast}\)
*is bounded below on*
\(H^{2}\)
*or*
\(A^{2}_{\alpha}\)
*if and only if*
\(\varphi\in\operatorname{Aut}(\mathbb {D})\)
*and*
\(\psi\in H^{\infty}\)
*is bounded away from zero on*
\(\mathbb {D}\).

## 3 The operators \(C_{\psi_{1},\varphi_{1}}C_{\psi_{2},\varphi _{2}}^{\ast}\) and \(C_{\psi_{1},\varphi_{1}}^{\ast}C_{\psi _{2},\varphi_{2}}\)

In this section, we find all Fredholm operators \(C_{\psi_{1},\varphi _{1}}C_{\psi_{2},\varphi_{2}}^{\ast}\) and \(C_{\psi_{1},\varphi _{1}}^{\ast}C_{\psi_{2},\varphi_{2}}\).

A linear-fractional self-map of \(\mathbb{D}\) is a mapping of the form \(\varphi(z)=(az+b)/(cz+d)\) with \(ad-bc \neq0\) such that \(\varphi (\mathbb{D}) \subseteq\mathbb{D}\). We denote the set of those maps by \(\operatorname{LFT}(\mathbb{D})\). Suppose \(\varphi(z)=(az+b)/(cz+d)\). It is well known that the adjoint of \(C_{\varphi}\) acting on \(H^{2}\) and \(A^{2}_{\alpha}\) is given by \(C_{\varphi}^{\ast}=T_{g}C_{\sigma}T_{h}^{\ast}\), where \(\sigma(z)=({\overline{a}z-\overline{c}})/({-\overline {b}z+\overline{d}})\) is a self-map of \(\mathbb{D}\), \(g(z)=(-\overline{b}z+\overline {d})^{-\gamma}\), and \(h(z)=(cz+d)^{\gamma}\). Note that *g* and *h* are in \(H^{\infty}\) ([17, Theorem 9.2]). If \(\varphi(\zeta)=\eta\) for \(\zeta,\eta\in\partial\mathbb{D}\), then \(\sigma(\eta)=\zeta\). We know that *φ* is an automorphism if and only if *σ* is, and in this case, \(\sigma=\varphi^{-1}\). The map *σ* is called the Krein adjoint of *φ*. We denote by \(F(\varphi)\) the set of all points in \(\partial\mathbb{D}\) at which *φ* has a finite angular derivative.

### Example 3.1

Suppose that \(\varphi\in\operatorname{LFT}(\mathbb{D})\) is not an automorphism of \(\mathbb{D}\). Assume that \(\psi\in H^{\infty}\) is continuously extendable to \(\mathbb{D} \cup F(\varphi )\). Assume that \(C_{\psi,\varphi}C_{\psi,\varphi}^{\ast}\) is considered as an operator on \(H^{2}\) or \(A_{\alpha}^{2}\). Since *φ* is not an automorphism of \(\mathbb{D}\), \(\overline {\varphi(\mathbb{D})} \subseteq\mathbb{D}\) or there is only one point \(\zeta\in\partial\mathbb{D}\) such that \(\varphi(\zeta) \in \partial\mathbb{D}\). If \(\overline{\varphi(\mathbb{D})} \subseteq \mathbb{D}\), then by [17, p. 129], \(C_{\varphi}\) is compact. It is easy to see that \(C_{\psi,\varphi}C_{\psi,\varphi}^{\ast}\) is a compact operator. Since compact operators are not Fredholm, we can see that \(C_{\psi,\varphi}C_{\psi,\varphi}^{\ast}\) is not Fredholm.

In the other case, assume that \(F(\varphi)=\{\zeta\}\). Because for each \(w \in\partial\mathbb{D}\) such that \(w \neq\zeta\), \(\sigma (\varphi(w)) \notin\partial\mathbb{D}\), we obtain \(\sigma \circ \varphi\notin\operatorname{Aut}(\mathbb{D})\). Since \(C_{\sigma\circ \varphi}\) is not Fredholm (see e.g. [2] and [16]), \(0 \in \sigma_{e}(C_{\sigma\circ\varphi})\). By [23, Corollary 2.2] and [4, Proposition 2.3] there is a compact operator *K* such that

Also, [23, Theorem 3.1], [23, Proposition 3.6], and [24, Theorem 3.2] imply that there is a compact operator \(K'\) such that

From the fact that \(0 \in\sigma_{e}(C_{\sigma\circ\varphi})\) and equation (2) we can infer that \(0 \in\sigma_{e}(C_{\psi,\varphi }C_{\psi,\varphi}^{\ast})\). Then \(C_{\psi,\varphi}C_{\psi,\varphi }^{\ast}\) is not Fredholm.

By the preceding example it seems natural to conjecture that if \(C_{\psi,\varphi}C_{\psi,\varphi}^{\ast}\) is Fredholm, then \(\varphi\in\operatorname{Aut}(\mathbb{D})\). We will prove our conjecture in Theorem 3.2 and show that if \(C_{\psi_{1},\varphi_{1}}C_{\psi _{2},\varphi_{2}}^{\ast}\) is Fredholm on \(H^{2}\) or \(A_{\alpha }^{2}\), then \(C_{\psi_{1},\varphi_{1}}\) and \(C_{\psi_{2},\varphi _{2}}\) are Fredholm.

### Theorem 3.2

*The operator*
\(C_{\psi_{1},\varphi_{1}}C_{\psi _{2},\varphi_{2}}^{\ast}\)
*is Fredholm on*
\(H^{2}\)
*or*
\(A_{\alpha}^{2}\)
*if and only if*
\(\varphi_{1},\varphi_{2} \in\operatorname{Aut}(\mathbb {D})\), \(\psi_{1},\psi_{2} \in H^{\infty}\), *and*
\(\psi_{1}\)
*and*
\(\psi _{2}\)
*are bounded away from zero near the unit circle*.

### Proof

Let \(C_{\psi_{1},\varphi_{1}}C_{\psi_{2},\varphi _{2}}^{\ast}\) be Fredholm. Therefore \(C_{\psi_{2},\varphi_{2}}^{\ast }\) is left semi-Fredholm. By Proposition 2.3 we see that \(\varphi_{2} \in\operatorname{Aut}(\mathbb{D})\) and \(\psi_{2} \in H^{\infty}\) is bounded away from zero near the unit circle. Since \(C_{\psi _{2},\varphi_{2}}C_{\psi_{1},\varphi_{1}}^{\ast}\) is Fredholm, again we can see that \(\varphi_{1}\) is an automorphism of \(\mathbb {D}\) and \(\psi_{1} \in H^{\infty}\) is bounded away from zero near the unit circle.

Conversely, let \(\varphi_{1},\varphi_{2} \in\operatorname{Aut}(\mathbb {D})\) and \(\psi_{1},\psi_{2} \in H^{\infty}\) be bounded away from zero near the unit circle. By Proposition 2.3, \(C_{\psi_{1},\varphi _{1}}\) and \(C_{\psi_{2},\varphi_{2}}^{\ast}\) are Fredholm, so the result follows. □

In the following theorem for functions \(\psi_{1},\psi_{2} \in A(\mathbb{D})\), we find all Fredholm operators \(C_{\psi_{1},\varphi _{1}}^{\ast}C_{\psi_{2},\varphi_{2}}\) when \(\varphi_{1}\) and \(\varphi_{2}\) are univalent self-maps of \(\mathbb{D}\).

### Theorem 3.3

*Suppose that*
\(\psi_{1},\psi_{2} \in A(\mathbb{D})\). *Let*
\(\varphi_{1}\)
*and*
\(\varphi_{2}\)
*be univalent self*-*maps of*
\(\mathbb{D}\). *The operator*
\(C_{\psi_{1},\varphi _{1}}^{\ast}C_{\psi_{2},\varphi_{2}}\)
*is Fredholm on*
\(H^{2}\)
*or*
\(A_{\alpha}^{2}\)
*if and only if*
\(C_{\psi_{1},\varphi_{1}}\)
*and*
\(C_{\psi_{2},\varphi_{2}}\)
*are Fredholm on*
\(H^{2}\)
*or*
\(A_{\alpha }^{2}\), *respectively*.

### Proof

Let \(C_{\psi_{1},\varphi_{1}}^{\ast}C_{\psi _{2},\varphi_{2}}\) be Fredholm on \(H^{2}\) or \(A_{\alpha}^{2}\). Then \(C_{\psi_{2},\varphi_{2}}^{\ast}C_{\psi_{1},\varphi_{1}}\) is also Fredholm. It is easy to see that \(C_{\varphi_{2}}\) and \(C_{\varphi _{1}}\) are left semi-Fredholm. Therefore, \(0 \notin\sigma _{\mathrm{le}}(C_{\varphi_{1}})\) and \(0 \notin\sigma_{\mathrm{le}}(C_{\varphi _{2}})\). Since \(\operatorname{dim} \operatorname{ker} C_{\psi_{1},\varphi _{1}}^{\ast}C_{\psi_{2},\varphi_{2}} < \infty\) and \(\operatorname{dim} \operatorname{ker} C_{\psi_{2},\varphi_{2}}^{\ast}C_{\psi _{1},\varphi_{1}} < \infty\), \(\psi_{1} \not\equiv0\), \(\psi_{2} \not\equiv0\), and \(\varphi _{1}\) and \(\varphi_{2}\) are not constant functions. By the open mapping theorem we know that \(0 \notin\sigma_{p}(C_{\varphi_{1}})\) and \(0 \notin\sigma_{p}(C_{\varphi_{2}})\). Now [18, Proposition 4.4, p. 359] implies that \(0 \notin\sigma_{\mathrm{ap}}(C_{\varphi _{1}})\) and \(0 \notin\sigma_{\mathrm{ap}}(C_{\varphi_{2}})\). Hence by [18, Proposition 6.4, p. 208], \(\operatorname{ran} C_{\varphi_{1}}\) and \(\operatorname{ran} C_{\varphi_{2}}\) are closed. By [1, Theorem 5.1], \(\varphi_{1},\varphi_{2} \in\operatorname{Aut}(\mathbb{D})\). Since \(C_{\varphi_{1}}^{\ast}\) and \(C_{\varphi_{2}}\) are invertible, \((C_{\varphi_{1}}^{\ast})^{-1}\) and \(C_{\varphi_{2}}^{-1}\) are Fredholm. Then \(T_{\psi_{1}}^{\ast}T_{\psi_{2}}\) is Fredholm, and so \(0 \notin\sigma_{e}(T_{\overline{\psi_{1}}\psi_{2}})\). We get from [25] and [20, Theorem 2] that \(\psi_{1}\) and \(\psi _{2}\) never vanish on \(\partial\mathbb{D}\). Since \(\psi_{1} \not \equiv0\) and \(\psi_{2} \not\equiv0\), \(\psi_{1}\) and \(\psi_{2}\) have only finite zeroes on \(\mathbb{D}\). This implies that there is \(r < 1\) such that for each *w* with \(r < |w| < 1\), \(\psi_{1}(w) \neq0\) and \(\psi_{2}(w) \neq0\). Therefore, \(\psi_{1}\) and \(\psi_{2}\) are bounded away from zero near the unit circle. Therefore, by Proposition 2.3, \(C_{\psi_{1},\varphi_{1}}\) and \(C_{\psi_{2},\varphi_{2}}\) are Fredholm on \(H^{2}\) or \(A_{\alpha}^{2}\). The reverse implication follows from the fact stated before Theorem 3.2. □

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Haji Shaabani, M. Fredholmness of multiplication of a weighted composition operator with its adjoint on \(H^{2}\) and \(A_{\alpha}^{2}\).
*J Inequal Appl* **2018**, 23 (2018). https://doi.org/10.1186/s13660-018-1615-0

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DOI: https://doi.org/10.1186/s13660-018-1615-0