Isometric composition operators on weighted Dirichlet-type spaces

Abstract

In this article, we characterize the surjective isometric composition operator C _φ on the weighted Dirichlet-type spaces of the unit disk $D$, where φ is an analytic self-map of $D$, and show that C _φ is a surjective isometry if and only if φ is a rotation map.

2000 MSC: Primary: 47B33; Secondary: 30D45; 30H30; 30H05; 47B38.

1. Introduction

The Lebesgue area measure on the unit disk $D$ in the complex plane is defined by dA(z) = rdrdt = dxdy. Denote by $H\left(D\right)$ the class of all analytic functions on $D$ and $S\left(D\right)$ the collection of all the analytic self mappings of $D$.

Given real numbers p > 0 and α > -1, the weighted Bergman space $A_{\alpha }^p$ is defined as the space of f in $H\left(D\right)$ such that

$${∥f∥}_{A_{\alpha }^p}={\left(\underset{D}{\int }{\left|f\left(z\right)\right|}^p{\left(1-{\left|z\right|}^2\right)}^{\alpha }\left(\alpha +1\right)dA\right)}^{1/p}<\infty .$$

The weighted Dirichlet-type space ${\mathbb{D}}_{\alpha }^p$ is the space of f in $H\left(D\right)$ such that $f^{\prime }\in A_{\alpha }^p$, equipped with the "norm":

$${∥f∥}_{{\mathcal{D}}_{\alpha }^p}=\left|f\left(0\right)\right|+{∥f^{\prime }∥}_{A_{\alpha }^p}.$$

It is not a true norm for 0 <p < 1, but it satisfies

$${∥f+g∥}_{{\mathcal{D}}_{\alpha }^p}\le C_p\left({∥f∥}_{{\mathcal{D}}_{\alpha }^p}+{∥g∥}_{{\mathcal{D}}_{\alpha }^p}\right),$$

where the constant C _p depends only on p. We write ${\mathbb{D}}^p={\mathbb{D}}_0^p$. The space $\mathbb{D}={\mathbb{D}}^2$ is the classical Dirichlet space of analytic functions whose image Riemann surface has finite area. Clearly, ${\mathbb{D}}^p\subset {\mathbb{D}}^q$ when q <p < ∞.

For $u\in H\left(D\right)$ and $\phi \in S\left(D\right)$, the composition operator C _φ induced by φ is defined as C _φ f = f ○ φ for $f\in H\left(D\right)$; the multiplication operator M _u induced by u is defined by M _u f(z) = u(z)f(z); and the weighted composition operator W_u,φinduced by φ and u is defined by (W_u,φf)(z) = u(z)f(φ(z)) for z ∈ Ω and f ∈ H(Ω). If we let u ≡ 1, then W _u,φ = C _φ ; if we let φ = id, then W _u,φ = M _u . So we can regard weighted composition operator as a generalization of a multiplication operator and a composition operator. These operators are linear. In [1], Hibschweiler studied the boundedness and compactness of composition operators on ${\mathbb{D}}^p$. In [2], Roan studied the boundness of composition operators on S^p, where $S^p=\left\{f\in H\left(D\right):f^{\prime }\in H^p\right\}$.

An operator T on a normed space X is said to be an isometric operator if ||Tf|| _X = ||f|| _X , for any f ∈ X.

The isometric composition operator on analytic functions spaces has been studied by many authors. In [3], Martín and Vukotić studied the isometric composition operators on H^p, $A_{\alpha }^p$ (see also Kolaski [4, 5]) for 1 ≤ p < ∞ and the analytic Besov spaces B^pfor 1 <p < ∞. They obtained that C _φ is an isometry of H^pif and only if φ is inner and φ(0) = 0, and C _φ is an isometry of $A_{\alpha }^p$ if and only if φ is a rotation. Also Carswell and Hammond [6] obtained that $C_{\phi }:A_{\alpha }^2\to A_{\alpha }^2$ is an isometry if and only if φ is a rotation; this fact differs somewhat from the analogs results that are known for other Hilbert spaces.

The isometric composition operators on the Bloch spaces in the unit disk were discussed by Martín and Vukotić [7], Colonna [8], Allen and Colonna [9, 10], Li and Zhou [11]. The same problems were studied on the Bloch spaces in the unit polydisk by Cohen and Colonna [12], in unit ball by Li [13], and Li and Ruan [14]. For the BMOA space, see [15]. In [16], Martín and Vukotić also studied the isometric composition operators on the classical Dirichlet spaces in the unit disk. They obtained that C _φ is an isometric operator if and only if φ is a univalent map of the unit disk such that $A\left[D\backslash \phi \left(D\right)\right]=0$ and φ(0) = 0. In [17], Novinger and Oberlin studied the isometric composition operators on S^pwith different norms.

The present article continues this line of research and discusses the isometric composition operators on the weighted Dirichlet-type space in the unit disk.

2. Main results

Our proofs will depend upon a characterization of the linear isometries of weighted Bergman spaces due to Kolaski [4]. Although not stated by Kolaski explicitly, the following result is a direct consequence of Theorems 1 and 4 in [4], which are much more general results than we will need here.

Theorem 2.1. Let 0 <p < ∞, p ≠ 2, and α > -1. Then every linear isometry$T:A_{\alpha }^p\to A_{\alpha }^p$takes the form (Tf)(z) = g(z) · f(φ(z)), for all$f\in A_{\alpha }^p$and$z\in D$, for some function φ which maps$D$conformally onto a dense subset of$D$, and where g = T 1. Moreover, if T is surjective, then φ is a disk automorphism and g = λ(φ')^(2+α)/pfor some |λ| = 1.

Following the ideas of Theorem 4.5.1 in [18], we investigate the surjective isometric composition operators on the weighted Dirichlet-type space.

Theorem 2.2. Let φ be a self-map of the unit disk. Then the induced composition operator C _φ is a surjective isometry on the weighted Dirichlet-type space${\mathbb{D}}_{\alpha }^p$, 1 ≤ p < ∞, p ≠ 2, -1 <α < ∞ if and only if φ is a rotation map.

Proof. Since the composition operator induced by a rotation is clearly an isometry, it suffices to show that if C _φ is a surjective isometry, then φ is a rotation.

Suppose that C _φ is a surjective isometric composition operator. Let n be a positive integer and t be a real number. We define the function p _n (z) = zⁿfor all $z\in D$. The weighted Dirichlet-type space contains the polynomials and thus $1+t{p}_n\in {\mathbb{D}}_{\alpha }^p$ for every real number t and positive integer n.

Using the definition of norm of ${\mathbb{D}}_{\alpha }^p$ and that C _φ is an isometry, we have

$${∥C_{\phi }1∥}_{{\mathcal{D}}_{\alpha }^p}={∥1∥}_{{\mathcal{D}}_{\alpha }^p},{∥C_{\phi }{p}_n∥}_{{\mathcal{D}}_{\alpha }^p}={∥p_n∥}_{{\mathcal{D}}_{\alpha }^p}$$

and

$${∥C_{\phi }\left(1+t{p}_n\right)∥}_{{\mathcal{D}}_{\alpha }^p}={∥1+t{p}_n∥}_{{\mathcal{D}}_{\alpha }^0}.$$

It follows that

$$\begin{array}{ll}\hfill {∥1+t{p}_n∥}_{{\mathcal{D}}_{\alpha }^p}& =1+{\left(\underset{D}{\int }{\left|{\left(t{p}_n\right)}^{\prime }\right|}^p{\left(1-{\left|z\right|}^2\right)}^{\alpha }\left(\alpha +1\right)dA\right)}^{\frac{1}{p}}\\ ={∥1∥}_{{\mathcal{D}}_{\alpha }^p}+{∥t{p}_n∥}_{{\mathcal{D}}_{\alpha }^p}={∥1∥}_{{\mathcal{D}}_{\alpha }^p}+\left|t\right|{∥p_n∥}_{{\mathcal{D}}_{\alpha }^p}\\ ={∥C_{\phi }1∥}_{{\mathcal{D}}_{\alpha }^p}+\left|t\right|{∥C_{\phi }{p}_n∥}_{{\mathcal{D}}_{\alpha }^p}\\ =\left|C_{\phi }1\left(0\right)\right|+\left|t\right|\left|C_{\phi }{p}_n\left(0\right)\right|+{∥{\left(C_{\phi }1\right)}^{\prime }∥}_{A_{\alpha }^p}+\left|t\right|{∥{\left(C_{\phi }{p}_n\right)}^{\prime }∥}_{A_{\alpha }^p}\end{array}$$

and

$${∥C_{\phi }\left(1+t{p}_n\right)∥}_{{\mathcal{D}}_{\alpha }^p}=\left|C_{\phi }1\left(0\right)+t{C}_{\phi }{p}_n\left(0\right)\right|+{∥{\left(C_{\phi }1\right)}^{\prime }+t{\left(C_{\phi }{p}_n\right)}^{\prime }∥}_{A_{\alpha }^p}.$$

So

$$\begin{array}{c}\left|C_{\phi }1\left(0\right)+t{C}_{\phi }{p}_n\left(0\right)\right|+{∥{\left(C_{\phi }1\right)}^{\prime }+t{\left(C_{\phi }{p}_n\right)}^{\prime }∥}_{A_{\alpha }^p}\\ =\left|C_{\phi }1\left(0\right)\right|+\left|t\right|\left|C_{\phi }{p}_n\left(0\right)\right|+{∥{\left(C_{\phi }1\right)}^{\prime }∥}_{A_{\alpha }^p}+\left|t\right|{∥{\left(C_{\phi }{p}_n\right)}^{\prime }∥}_{A_{\alpha }^p}.\end{array}$$

A simple application of the triangle inequality for the norms shows that

$$\left|C_{\phi }1\left(0\right)+t{C}_{\phi }{p}_n\left(0\right)\right|\le \left|C_{\phi }1\left(0\right)\right|+\left|t\right|\left|C_{\phi }{p}_n\left(0\right)\right|$$

and

$${∥{\left(C_{\phi }1\right)}^{\prime }+t{\left(C_{\phi }{p}_n\right)}^{\prime }∥}_{A_{\alpha }^p}\le {∥{\left(C_{\phi }1\right)}^{\prime }∥}_{A_{\alpha }^p}+\left|t\right|{∥{\left(C_{\phi }{p}_n\right)}^{\prime }∥}_{A_{\alpha }^p}.$$

Consequently,

$$\left|C_{\phi }1\left(0\right)+t{C}_{\phi }{p}_n\left(0\right)\right|=\left|C_{\phi }1\left(0\right)\right|+\left|t\right|\left|C_{\phi }{p}_n\left(0\right)\right|$$

and

$${∥{\left(C_{\phi }1\right)}^{\prime }+t{\left(C_{\phi }{p}_n\right)}^{\prime }∥}_{A_{\alpha }^p}={∥{\left(C_{\phi }1\right)}^{\prime }∥}_{A_{\alpha }^p}+\left|t\right|{∥{\left(C_{\phi }{p}_n\right)}^{\prime }∥}_{A_{\alpha }^p}.$$

In particular, for n = 1, we get |1 + tφ(0)| = 1 + |t||φ(0)| which implies, since t is an arbitrary real number, that φ(0) = 0.

Therefore ${∥C_{\phi }f∥}_{{\mathcal{D}}_{\alpha }^p}={∥f∥}_{{\mathcal{D}}_{\alpha }^p}$ is equivalent to ${∥{\left(C_{\phi }f\right)}^{\prime }∥}_{A_{\alpha }^p}={∥f^{\prime }∥}_{A_{\alpha }^p}$.

Let ${\mathbb{D}}_{\alpha ,0}^p$ denotes the subspace of functions in ${\mathbb{D}}_{\alpha }^p$ that vanish at the origin. The differentiation operator D maps ${\mathbb{D}}_{\alpha ,0}^p$ isometrically onto $A_{\alpha }^p$ and its inverse I is given by

$$Ig\left(z\right)=\underset{0}{\overset{z}{\int }}g\left(\xi \right)d\xi$$

and maps $A_{\alpha }^p$ isometrically onto ${\mathbb{D}}_{\alpha ,0}^p$.

Since C _φ is a surjective isometry on ${\mathbb{D}}_{\alpha }^p$, for every $f\in {\mathbb{D}}_{\alpha ,0}^p$, there exists $g\in {\mathbb{D}}_{\alpha }^p$ such that C _φ g = f. Because

$$0=f\left(0\right)=\left(C_{\phi }g\right)\left(0\right)=g\left(\phi \left(0\right)\right)=g\left(0\right),$$

then $g\in {\mathbb{D}}_{\alpha }^p$. Thus the isometric composition operator C _φ maps the subspace ${\mathbb{D}}_{\alpha ,0}^p$ onto itself. So the composition $D{C}_{\phi }I:A_{\alpha }^p\to A_{\alpha }^p$ is a surjective isometric operator. Set T = DC _φ I. Then by Theorem 2.1, there exists an automorphism φ of $D$ such that Tf(z) = g(z)f(φ(z)), for all $f\in A_{\alpha }^p$, $z\in D$, where g = T 1. Then, noting that I(f') = f - f(0), we obtain

$$g\left(z\right)f^{\prime }\left(\phi \left(z\right)\right)=T{f}^{\prime }\left(z\right)=D{C}_{\phi }I{f}^{\prime }\left(z\right)=D{C}_{\phi }\left(f\left(z\right)-f\left(0\right)\right)=f^{\prime }\left(\phi \left(z\right)\right){\phi }^{\prime }\left(z\right).$$

Therefore,

$$f^{\prime }\left(\phi \left(z\right)\right){\phi }^{\prime }\left(z\right)=g\left(z\right)f^{\prime }\left(\phi \left(z\right)\right)$$

(2.1)

for any $f\in {\mathbb{D}}_{\alpha }^p$.

Letting f = id in (2.1), then φ'(z) = g(z) (Alternatively, one can see that φ' = g since g(z) = (T 1)(z) = DC _φ I₁(z) = φ'(z)). Letting f = p₂, then φ(z)φ'(z) = g(z)φ(z), so from g(z) ≠ 0, we have φ(z) = φ(z). Since φ is an automorphism of $D$ and φ(0) = 0, it follows that φ(z) = λz, where |λ| = 1, that is φ is a rotation.

From the proof of the above theorem, it is easy to see that if C _φ is an isometry on ${\mathbb{D}}_{\alpha }^p$, 1 ≤ p < ∞ then φ(0) = 0. We can get the following corollary.

Corollary 2.3. Let$\phi \in \text{Aut}\left(D\right)$. Then the induced composition operator C _φ is an isometry on the weighted Dirichlet-type space${\mathbb{D}}_{\alpha }^p$, 1 ≤ p < ∞ and -1 <α < ∞, if and only if φ is a rotation map.

Clearly, if both C _φ and M _u are isometries, then W_u,φis an isometry, the following theorem will show that the converse is also true.

Theorem 2.4. The weighted composition operator W _u,φ is an isometric operator on${\mathbb{D}}_{\alpha }^p$, 1 ≤ p < ∞ and α > -1, if and only if both C _φ and M _u are isometric operators.

Proof. Suppose W _u,φ is an isometry. Replacing C _φ by W _u,φ and using the same methods and the same assumption in Theorem 2.2, we get

$${∥{\left(W_{u,\phi }1\right)}^{\prime }+t{\left(W_{u,\phi }{p}_n\right)}^{\prime }∥}_{A_{\alpha }^p}={∥{\left(W_{u,\phi }1\right)}^{\prime }∥}_{A_{\alpha }^p}+\left|t\right|{∥{\left(W_{u,\phi }{p}_n\right)}^{\prime }∥}_{A_{\alpha }^p}$$

(2.2)

It follows from (2.2) that the real valued function

$$p\left(t\right)={∥{\left(W_{u,\phi }1\right)}^{\prime }+t{\left(W_{u,\phi }{p}_n\right)}^{\prime }∥}_{A_{\alpha }^p}$$

is not a differentiable function of t at t = 0. However, in the terminology of [19], the L^pnorm is weakly differentiable at every point except the zero vector, that is ${∥\cdot ∥}_{A_{\alpha }^p}$ is not weakly differentiable at (W _u,φ 1)'. Consequently

$${\left(W_{u,\phi }1\right)}^{\prime }\left(z\right)=0$$

for every $z\in D$. So (W _u,φ 1)' ≡ 0, which implies that u = W _u,φ 1 is a constant.

Since W _u,φ is an isometry, ${∥u∥}_{{\mathcal{D}}_{\alpha }^p}={∥W_{u,\phi }1∥}_{{\mathcal{D}}_{\alpha }^p}={∥1∥}_{{\mathcal{D}}_{\alpha }^p}=1$, and consequently, u ≡ λ, for some |λ| = 1. Hence the multiplication operator M _u is an isometry. Now,

$${∥f∥}_{{\mathcal{D}}_{\alpha }^p}={∥W_{u,\phi }f∥}_{{\mathcal{D}}_{\alpha }^p}={∥\left(u{C}_{\phi }\right)\left(f\right)∥}_{{\mathcal{D}}_{\alpha }^p}={∥\lambda C_{\phi }\left(f\right)∥}_{{\mathcal{D}}_{\alpha }^p}={∥C_{\phi }\left(f\right)∥}_{{\mathcal{D}}_{\alpha }^p}$$

for every $f\in {\mathbb{D}}_{\alpha }^p$. Hence the composition operator C _φ is an isometry.

Remark. From the above the theorem, we can get that the multiplier operator M _u is an isometric operator on ${\mathbb{D}}_{\alpha }^p$, 1 ≤ p < ∞, -1 <α < ∞ if and only if u is a constant of modulus one. In [20], Aleman et al. characterized the nontrivial isometric multipliers on the Dirichlet-type space ${\mathbb{D}}_w^p$, 1 ≤ p < ∞ (here w is the weighted function). For the sake of completeness, we state their results as follows.

Theorem 2.5. Let 1 ≤ p < ∞. Suppose that the Dirichlet-type space${\mathbb{D}}_w^p$is complete and that point-evaluations are bounded. Then${\mathbb{D}}_w^p$has nonconstant isometric pointwise multipliers if and only if p = 2 and w(z) = -2log|z| a.e. in$D$. In this case${\mathbb{D}}_w^p=H^2$and the isometric multipliers are precisely the inner functions.