Isometries on Products of Composition and Integral Operators on Bloch Type Space

Abstract

We characterize the isometries on the products of composition and integral operators on the Bloch type space in the disk.

1. Introduction

Let be the unit disk of the complex plane, and be the set of analytic self-maps of . The algebra of all holomorphic functions with domain will be denoted by .

We recall that the Bloch type space    consists of all such that

(1.1)

then is a complete seminorm on , which is Möbius invariant.

It is well known that is a Banach space under the norm

(1.2)

Let be an analytic self-map of , then the composition operator induced by is defined by

(1.3)

for and

Let , then the integer operator is defined by

(1.4)

for .

The products of composition and integral type operators were first introduced and discussed by Li and Stević [1–3], which are defined by

(1.5)

Let and be two Banach spaces, recall that a linear isometry is a linear operator from to such that for all .

In [4], Banach raised the question concerning the form of an isometry on a specific Banach space. In most cases, the isometries of a space of analytic functions on the disk or the ball have the canonical form of weighted composition operators, which is also true for most symmetric function spaces. For example, the surjective isometries of Hardy and Bergman spaces are certain weighted composition operators (see [5–7]).

The description of all isometric composition operators is known for the Hardy space (see [8]). An analogous statement for the Bergman space with standard radial weights has recently been obtained in [9], and there is a unified proof for all Hardy spaces and also for arbitrary Bergman spaces with reasonable radial weights [10]. For the Dirichlet space and Bloch space, the reader is referred to [11, 12], and for the BMOA, see [13].

The surjective isometries of the Bloch space are characterized in [14]. Trivially, every rotation induces an isometry of . It has recently been shown in [15] that for composition operators, which induce isometries of , the conditions and must hold. Here, denotes the (global) cluster set of , that is, the set of all points such that there exists a sequence in with the properties and as . Plenty of information on cluster sets is contained in [16].

Continued the work, in 2008, Bonet et al. [17] discussed isometric weighted composition operators on weighted Banach spaces of type In 2008, Cohen and Colonna [18] discussed the spectrum of an isometric composition operators on the Bloch space of the polydisk. In 2009, Allen and Colonna [19] investigated the isometric composition operators on the Bloch space in . They [20] also discussed the isometries and spectra of multiplication operators on the Bloch space in the disk. Isometries of weighted spaces of holomorphic functions on unbounded domains were discussed by Boyd and Rueda in [21].

Building on those foundations, the present paper continues this line of research, and discusses the isometries on the products of composition and integral operators on the Bloch type space in the disk.

2. Notations and Lemmas

To begin the discussion, let us introduce some notations and state a couple of lemmas.

For , the involution which interchanges the origin and point , is defined by

(2.1)

For , in , the pseudohyperbolic distance between and is given by

(2.2)

and the hyperbolic metric is given by

(2.3)

where is any piecewise smooth curve in from to .

The following lemma is well known [22].

Lemma 2.1.

For all , one has

(2.4)

For , the Schwarz-Pick lemma shows that , and if the equality holds for some , then is an automorphism of the disk. It is also well known that, for , is always bounded on .

A little modification of Lemma in [17] shows the following lemma.

Lemma 2.2.

There exists a constant such that

(2.5)

for all and .

Throughout the remainder of this paper, will denote a positive constant, the exact value of which will vary from one appearance to the next.

3. Main Theorems

Theorem 3.1.

Let be analytic self-maps of the unit disk and then, the operator is an isometry in the seminorm if and only if the following conditions hold.

(A);

(B) For every , there exists at least a sequence in , such that and

Proof.

We prove the sufficiency first.

By condition , for every we have

(3.1)

Next we show that property (B) implies

In fact, given any then for some sequence For any fixed , it follows from (B) that there is a sequence such that

(3.2)

as By Lemma 2.2, for all and ,

(3.3)

Hence,

(3.4)

Therefore,

(3.5)

The inequality follows by letting

From the above discussions, we have , which means that is an isometry operator on the Bloch type space in the seminorm.

Now we turn to the necessity.

For any , we begin by taking test function

(3.6)

It is clear that Using Lemma 2.1, we have

(3.7)

So,

(3.8)

On the other hand, since we have By isometry assumption, for any , we have

(3.9)

So (A) follows by noticing is arbitrary.

Since there exists a sequence such that

(3.10)

as

It follows from (A) that

(3.11)

(3.12)

Combining (3.10) and (3.12), it follows that

(3.13)

The last inequality follows (3.7) since .

Consequently,

(3.14)

That is,

Combining (3.10), (3.11), and (3.14), we know that

(3.15)

This completes the proof of this theorem.

Theorem 3.2.

Let and be an analytic self-map of the unit disk, such that fixes the origin, then the operator is an isometry in the seminorm if and only if the following conditions hold.

(C);

(D)For every , there exists at least a sequence in , such that and .

Proof.

We prove the sufficiency first.

By condition (C), for every we have

(3.16)

Next we show that the property (D) implies

In fact, given any then for some sequence For any fixed , by property (D), there is a sequence such that

(3.17)

as By Lemma 2.2, for all and ,

(3.18)

Hence,

(3.19)

Therefore,

(3.20)

The inequality follows by letting

Now we turn to the necessity.

For any , we use the same test function defined by (3.6) which satisfies By isometry assumption, for any , we have

(3.21)

So, (C) follows by noticing is arbitrary.

Since thus there exists a sequence such that

(3.22)

as

It follows from (C) that

(3.23)

(3.24)

Combining (3.22) and (3.24), it follows that

(3.25)

The last inequality follows (3.7), since .

Consequently,

(3.26)

That is, .

Combining (3.22), (3.23), and (3.26), we know that

(3.27)

the desired results follows. The proof of this theorem is completed.

Remark 3.3.

If , then , will be Bloch space , so the similar results on the Bloch space corresponding to Theorems 3.1 and 3.2 also hold.