- Research Article
- Open Access
Isometries on Products of Composition and Integral Operators on Bloch Type Space
© Geng-Lei Li and Ze-Hua Zhou. 2010
- Received: 8 June 2010
- Accepted: 12 July 2010
- Published: 26 July 2010
We characterize the isometries on the products of composition and integral operators on the Bloch type space in the disk.
- Banach Space
- Integral Operator
- Hardy Space
- Composition Operator
- Bergman Space
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 .
then is a complete seminorm on , which is Möbius invariant.
Let and be two Banach spaces, recall that a linear isometry is a linear operator from to such that for all .
In , 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 ). An analogous statement for the Bergman space with standard radial weights has recently been obtained in , and there is a unified proof for all Hardy spaces and also for arbitrary Bergman spaces with reasonable radial weights . For the Dirichlet space and Bloch space, the reader is referred to [11, 12], and for the BMOA, see .
The surjective isometries of the Bloch space are characterized in . Trivially, every rotation induces an isometry of . It has recently been shown in  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 .
Continued the work, in 2008, Bonet et al.  discussed isometric weighted composition operators on weighted Banach spaces of type In 2008, Cohen and Colonna  discussed the spectrum of an isometric composition operators on the Bloch space of the polydisk. In 2009, Allen and Colonna  investigated the isometric composition operators on the Bloch space in . They  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 .
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.
To begin the discussion, let us introduce some notations and state a couple of lemmas.
where is any piecewise smooth curve in from to .
The following lemma is well known .
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  shows the following lemma.
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.
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.
(B) For every , there exists at least a sequence in , such that and
We prove the sufficiency first.
Next we show that property (B) implies
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.
So (A) follows by noticing is arbitrary.
The last inequality follows (3.7) since .
This completes the proof of this theorem.
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.
(D)For every , there exists at least a sequence in , such that and .
We prove the sufficiency first.
Next we show that the property (D) implies
The inequality follows by letting
Now we turn to the necessity.
So, (C) follows by noticing is arbitrary.
The last inequality follows (3.7), since .
That is, .
the desired results follows. The proof of this theorem is completed.
If , then , will be Bloch space , so the similar results on the Bloch space corresponding to Theorems 3.1 and 3.2 also hold.
This work was partially supported by the National Natural Science Foundation of China (Grant nos. 10971153, 10671141).
- Li S, Stević S: Products of composition and integral type operators from to the Bloch space. Complex Variables and Elliptic Equations 2008, 53(5):463–474. 10.1080/17476930701754118MathSciNetView ArticleMATHGoogle Scholar
- Li S, Stević 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):40–52. 10.1016/j.jmaa.2008.03.063MathSciNetView ArticleMATHGoogle Scholar
- Li S, Stević S: Products of integral-type operators and composition operators between Bloch-type spaces. Journal of Mathematical Analysis and Applications 2009, 349(2):596–610. 10.1016/j.jmaa.2008.09.014MathSciNetView ArticleMATHGoogle Scholar
- Banach S: Theorie des Operations Lineares. Chelsea, Warzaw, Poland; 1932.MATHGoogle Scholar
- Kolaski CJ: Isometries of weighted Bergman spaces. Canadian Journal of Mathematics 1982, 34(4):910–915. 10.4153/CJM-1982-063-5MathSciNetView ArticleMATHGoogle Scholar
- Kolaski CJ: Isometries of some smooth normed spaces of analytic functions. Complex Variables. Theory and Application 1988, 10(2–3):115–122.MathSciNetView ArticleMATHGoogle Scholar
- Korányi A, Vági S: Isometries of spaces of bounded symmetric domains. Canadian Journal of Mathematics 1976, 28(2):334–340. 10.4153/CJM-1976-035-8MathSciNetView ArticleMATHGoogle Scholar
- Cload BA: Composition operators: hyperinvariant subspaces, quasi-normals and isometries. Proceedings of the American Mathematical Society 1999, 127(6):1697–1703. 10.1090/S0002-9939-99-04663-8MathSciNetView ArticleMATHGoogle Scholar
- Carswell BJ, Hammond C: Composition operators with maximal norm on weighted Bergman spaces. Proceedings of the American Mathematical Society 2006, 134(9):2599–2605. 10.1090/S0002-9939-06-08271-2MathSciNetView ArticleMATHGoogle Scholar
- Martín MJ, Vukotić 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; 2006:133–138.View ArticleGoogle Scholar
- Martín MJ, Vukotić D: Isometries of the Dirichlet space among the composition operators. Proceedings of the American Mathematical Society 2006, 134(6):1701–1705. 10.1090/S0002-9939-05-08182-7MathSciNetView ArticleMATHGoogle Scholar
- Martín MJ, Vukotić D: Isometries of the Bloch space among the composition operators. Bulletin of the London Mathematical Society 2007, 39(1):151–155.MathSciNetView ArticleMATHGoogle Scholar
- Laitila J: Isometric composition operators on BMOA. to appear in Mathematische Nachrichten to appear in Mathematische NachrichtenGoogle Scholar
- Cima JA, Wogen WR: On isometries of the Bloch space. Illinois Journal of Mathematics 1980, 24(2):313–316.MathSciNetMATHGoogle Scholar
- Xiong C: Norm of composition operators on the Bloch space. Bulletin of the Australian Mathematical Society 2004, 70(2):293–299. 10.1017/S0004972700034511MathSciNetView ArticleMATHGoogle Scholar
- Collingwood 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.View ArticleGoogle Scholar
- Bonet J, Lindström M, Wolf E: Isometric weighted composition operators on weighted Banach spaces of type . Proceedings of the American Mathematical Society 2008, 136(12):4267–4273. 10.1090/S0002-9939-08-09631-7MathSciNetView ArticleMATHGoogle Scholar
- Cohen 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; 2008:9–21.View ArticleGoogle Scholar
- Allen RF, Colonna F: On the isometric composition operators on the Bloch space in . Journal of Mathematical Analysis and Applications 2009, 355(2):675–688. 10.1016/j.jmaa.2009.02.023MathSciNetView ArticleMATHGoogle Scholar
- Allen RF, Colonna F: Isometries and spectra of multiplication operators on the Bloch space. Bulletin of the Australian Mathematical Society 2009, 79(1):147–160. 10.1017/S0004972708001196MathSciNetView ArticleMATHGoogle Scholar
- Boyd 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):253–271. 10.1017/S0308210507001230MathSciNetView ArticleMATHGoogle Scholar
- Zhu K: Spaces of Holomorphic Functions in the Unit Ball, Graduate Texts in Mathematics. Volume 226. Springer, New York, NY, USA; 2005:x+271.MATHGoogle Scholar
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