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On the Isometric Composition Operators on the Bloch Space in Cn Robert F. Allen
 

Summary: On the Isometric Composition Operators on the Bloch Space in Cn
Robert F. Allen
George Mason University, Department of Mathematical Sciences
Fairfax, VA 22030, USA
Flavia Colonna
George Mason University, Department of Mathematics Sciences
Fairfax, VA 22030, USA
Abstract
Let be a holomorphic self-map of a bounded homogeneous domain D in Cn
. In this work, we show that
the composition operator C : f f is bounded on the Bloch space B of the domain and provide
estimates on its operator norm. We also give a sufficient condition for to induce an isometry on B. This
condition allows us to construct non-trivial examples of isometric composition operators in the case when D
has the unit disk as a factor. We then obtain some necessary conditions for C to be an isometry on B when
D is a Cartan classical domain. Finally, we give the complete description of the spectrum of the isometric
composition operators in the case of the unit disk and for a wide class of symbols on the polydisk.
Key words: Composition operators, Bloch space, Homogeneous domains, Isometry
2000 MSC: Primary: 30D45, 32M15; Secondary: 47B38,, 47A30
1. Introduction
An analytic function f on D = {z C : |z| < 1} is said to be Bloch if

  

Source: Allen, Robert F. - Mathematics Department, University of Wisconsin-La Crosse

 

Collections: Mathematics