Hypercyclic Behavior of Translation Operators on Spaces of Analytic Functions on Hilbert Spaces

Zoryana Mozhyrovska,Andriy V Zagorodnyuk

doi:10.1155/2015/139289

Abstract

We consider special Hilbert spaces of analytic functions of many infinite variables and examine composition operators on these spaces. In particular, we prove that under some conditions a translation operator is bounded and hypercyclic.

Highlights

As for first results related to hypercyclic operators there are classical works of Birkhoff [2] and MacLane [3] showing that the operators of translation and differentiation, acting on the space of entire functions of one complex variable, are hypercyclic
There are many results related to hypercyclic operators on spaces of analytic functions on finite and infinite-dimensional spaces
We examine the hypercyclic behavior of composition operators on Hilbert spaces of entire functions of many infinite variables

Summary

Introduction

There are many results related to hypercyclic operators on spaces of analytic functions on finite and infinite-dimensional spaces (see, e.g., [1, 4, 5]). Aron and Bes in [7] proved that the operator of composition with translation Ta is hypercyclic in the space of weakly continuous analytic functions on all bounded subsets of a separable Banach space X which are bounded on bounded subsets. Hypercyclic composition operators on spaces of analytic functions of finite and infinite many variables were studied in [8]. The purpose of this paper is to prove a generalization of the Chan and Shapiro’s result for Hilbert spaces of entire functions of infinite many variables. For background on analytic functions on Banach spaces we refer the reader to [12, 13]

Symmetric Fock Spaces and Analytic Functions

Hilbert Spaces of Entire Functions

Differentiation and Translation Operators on Hη