# Isometric composition operators on weighted Dirichlet-type spaces

- Li-Gang Geng
^{1}, - Ze-Hua Zhou
^{2}Email author and - Xing-Tang Dong
^{2}

**2012**:23

https://doi.org/10.1186/1029-242X-2012-23

© Geng et al; licensee Springer. 2012

**Received: **12 April 2011

**Accepted: **9 February 2012

**Published: **9 February 2012

## 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.

## Keywords

## 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 $D$.

*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*in $H\left(D\right)$ such that ${f}^{\prime}\in {A}_{\alpha}^{p}$, equipped with the "norm":

*p*< 1, but it satisfies

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,φ}induced by *φ* and *u* is defined by (*W*_{u,φ}*f*)(*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*^{
p
}for 1 <*p* < ∞. They obtained that *C*_{
φ
}is an isometry of *H*^{
p
}if 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*^{
p
}with 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)/p}*for 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*) = *z*^{
n
}for 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.*

*C*

_{ φ }is an isometry, we have

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 ${\u2225{C}_{\phi}f\u2225}_{{\mathcal{D}}_{\alpha}^{p}}={\u2225f\u2225}_{{\mathcal{D}}_{\alpha}^{p}}$ is equivalent to ${\u2225{\left({C}_{\phi}f\right)}^{\prime}\u2225}_{{A}_{\alpha}^{p}}={\u2225{f}^{\prime}\u2225}_{{A}_{\alpha}^{p}}$.

*D*maps ${\mathbb{D}}_{\alpha ,0}^{p}$ isometrically onto ${A}_{\alpha}^{p}$ and its inverse

*I*is given by

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

*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

*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

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
}*I*_{1}(*z*) = *φ*'(*z*)). Letting *f* = *p*_{2}, 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,φ}is 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

*t*at

*t*= 0. However, in the terminology of [19], the

*L*

^{ p }norm is weakly differentiable at every point except the zero vector, that is ${\u2225\cdot \u2225}_{{A}_{\alpha}^{p}}$ is not weakly differentiable at (

*W*

_{ u,φ }1)'. Consequently

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

*W*

_{ u,φ }is an isometry, ${\u2225u\u2225}_{{\mathcal{D}}_{\alpha}^{p}}={\u2225{W}_{u,\phi}1\u2225}_{{\mathcal{D}}_{\alpha}^{p}}={\u22251\u2225}_{{\mathcal{D}}_{\alpha}^{p}}=1$, and consequently,

*u*≡

*λ*, for some |

*λ*| = 1. Hence the multiplication operator

*M*

_{ u }is an isometry. Now,

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.*

## Declarations

### Acknowledgements

The authors would like to thank the referees for the useful comments and suggestions which improved the presentation of this article. ZHZ supported in part by the National Natural Science Foundation of China (Grant Nos. 10971153, 10671141).

## Authors’ Affiliations

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