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

*Journal of Inequalities and Applications*
**volumeÂ 2010**, ArticleÂ number:Â 184957 (2010)

## 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 â€‰â€‰ consists of all such that

then is a complete seminorm on , which is MÃ¶bius invariant.

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

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

for and

Let , then the integer operator is defined by

for .

The products of composition and integral type operators were first introduced and discussed by Li and SteviÄ‡ [1â€“3], which are defined by

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 [5â€“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

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

and the hyperbolic metric is given by

where is any piecewise smooth curve in from to .

The following lemma is well known [22].

Lemma 2.1.

For all , one has

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

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

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

as By Lemma 2.2, for all and ,

Hence,

Therefore,

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

It is clear that Using Lemma 2.1, we have

So,

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

So (A) follows by noticing is arbitrary.

Since there exists a sequence such that

as

It follows from (A) that

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

The last inequality follows (3.7) since .

Consequently,

That is,

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

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

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

as By Lemma 2.2, for all and ,

Hence,

Therefore,

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

So, (C) follows by noticing is arbitrary.

Since thus there exists a sequence such that

as

It follows from (C) that

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

The last inequality follows (3.7), since .

Consequently,

That is, .

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

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.

## References

Li S, SteviÄ‡ S: Products of composition and integral type operators from to the Bloch space.

*Complex Variables and Elliptic Equations*2008, 53(5):463â€“474. 10.1080/17476930701754118Li S, SteviÄ‡ S: Products of Volterra type operator and composition operator from and Bloch spaces to Zygmund spaces.

*Journal of Mathematical Analysis and Applications*2008, 345(1):40â€“52. 10.1016/j.jmaa.2008.03.063Li S, SteviÄ‡ S: Products of integral-type operators and composition operators between Bloch-type spaces.

*Journal of Mathematical Analysis and Applications*2009, 349(2):596â€“610. 10.1016/j.jmaa.2008.09.014Banach S:

*Theorie des Operations Lineares*. Chelsea, Warzaw, Poland; 1932.Kolaski CJ: Isometries of weighted Bergman spaces.

*Canadian Journal of Mathematics*1982, 34(4):910â€“915. 10.4153/CJM-1982-063-5Kolaski CJ: Isometries of some smooth normed spaces of analytic functions.

*Complex Variables. Theory and Application*1988, 10(2â€“3):115â€“122.KorÃ¡nyi A, VÃ¡gi S: Isometries of spaces of bounded symmetric domains.

*Canadian Journal of Mathematics*1976, 28(2):334â€“340. 10.4153/CJM-1976-035-8Cload BA: Composition operators: hyperinvariant subspaces, quasi-normals and isometries.

*Proceedings of the American Mathematical Society*1999, 127(6):1697â€“1703. 10.1090/S0002-9939-99-04663-8Carswell BJ, Hammond C: Composition operators with maximal norm on weighted Bergman spaces.

*Proceedings of the American Mathematical Society*2006, 134(9):2599â€“2605. 10.1090/S0002-9939-06-08271-2MartÃn MJ, VukotiÄ‡ D: Isometries of some classical function spaces among the composition operators. In

*Recent Advances in Operator-Related Function Theory, Contemporary Mathematics*.*Volume 393*. Edited by: Matheson AL, Stessin MI, Timoney RM. American Mathematical Society, Providence, RI, USA; 2006:133â€“138.MartÃn MJ, VukotiÄ‡ D: Isometries of the Dirichlet space among the composition operators.

*Proceedings of the American Mathematical Society*2006, 134(6):1701â€“1705. 10.1090/S0002-9939-05-08182-7MartÃn MJ, VukotiÄ‡ D: Isometries of the Bloch space among the composition operators.

*Bulletin of the London Mathematical Society*2007, 39(1):151â€“155.Laitila J: Isometric composition operators on BMOA. to appear in

*Mathematische Nachrichten*to appear in Mathematische NachrichtenCima JA, Wogen WR: On isometries of the Bloch space.

*Illinois Journal of Mathematics*1980, 24(2):313â€“316.Xiong C: Norm of composition operators on the Bloch space.

*Bulletin of the Australian Mathematical Society*2004, 70(2):293â€“299. 10.1017/S0004972700034511Collingwood EF, Lohwater AJ:

*The Theory of Cluster Sets, Cambridge Tracts in Mathematics and Mathematical Physics, no. 56*. Cambridge University Press, Cambridge, UK; 1966:xi+211.Bonet J, LindstrÃ¶m M, Wolf E: Isometric weighted composition operators on weighted Banach spaces of type .

*Proceedings of the American Mathematical Society*2008, 136(12):4267â€“4273. 10.1090/S0002-9939-08-09631-7Cohen J, Colonna F: Isometric composition operators on the Bloch space in the polydisk. In

*Banach Spaces of Analytic Functions, Contemporary Mathematics*.*Volume 454*. American Mathematical Society, Providence, RI, USA; 2008:9â€“21.Allen RF, Colonna F: On the isometric composition operators on the Bloch space in .

*Journal of Mathematical Analysis and Applications*2009, 355(2):675â€“688. 10.1016/j.jmaa.2009.02.023Allen RF, Colonna F: Isometries and spectra of multiplication operators on the Bloch space.

*Bulletin of the Australian Mathematical Society*2009, 79(1):147â€“160. 10.1017/S0004972708001196Boyd C, Rueda P: Isometries of weighted spaces of holomorphic functions on unbounded domains.

*Proceedings of the Royal Society of Edinburgh. Section A*2009, 139(2):253â€“271. 10.1017/S0308210507001230Zhu K:

*Spaces of Holomorphic Functions in the Unit Ball, Graduate Texts in Mathematics*.*Volume 226*. Springer, New York, NY, USA; 2005:x+271.

## Acknowledgment

This work was partially supported by the National Natural Science Foundation of China (Grant nos. 10971153, 10671141).

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Li, GL., Zhou, ZH. Isometries on Products of Composition and Integral Operators on Bloch Type Space.
*J Inequal Appl* **2010**, 184957 (2010). https://doi.org/10.1155/2010/184957

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DOI: https://doi.org/10.1155/2010/184957