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.
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.
2. Notations and Lemmas
To begin the discussion, let us introduce some notations and state a couple of lemmas.
The following lemma is well known .
A little modification of Lemma in  shows the following lemma.
3. Main Theorems
We prove the sufficiency first.
Now we turn to the necessity.
This completes the proof of this theorem.
We prove the sufficiency first.
Now we turn to the necessity.
the desired results follows. The proof of this theorem is completed.
This work was partially supported by the National Natural Science Foundation of China (Grant nos. 10971153, 10671141).
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