Non-wandering Operator in Bargmann Space

Minggang Wang,Hua Xu

doi:10.5539/jmr.v2n2p34

Abstract

In this paper the Bargmann space is denoted by F. This space’s roots can be found in mathematical problem of relativisticphysics or in quantum optics. In physics the Bargmann space contains the canonical coherent states, so it is the maintool for studying the bosonic coherent state theory of radiation field and for other application .This paper deals with theunilateral backward shift operator T on a Bargmann space F. We provide a sufficient condition for an unbounded operatorto be non-wandering operator, and then apply the condition to give a necessary and sufficient condition in order that T bea non-wandering operator.

Highlights

It is well known that linear operators in finite-dimensional linear spaces can’t be chaotic but the nonlinear operator may be
Lixin Tian and other researchers introduced non-wandering operators in infinite-dimensional Banach space, which are the generalization of Axiom A dynamic system but different from it
In physics the Bargmann space contains the canonical coherent states, so it is the main tool for studying the bosonic coherent state theory of radiation field

Summary

Introduction

It is well known that linear operators in finite-dimensional linear spaces can’t be chaotic but the nonlinear operator may be. Lixin Tian and other researchers introduced non-wandering operators in infinite-dimensional Banach space, which are the generalization of Axiom A dynamic system but different from it They are new linear chaotic operators and relative to hypercyclic operators, but different from them (Lixin Tian, 2005). In finite-dimensional separable Banach space, for the bounded linear operators, Lixin Tian and other researchers have given the definition of non-wandering operator (Lixin Tian, 2005). This definition is restricted for the bounded linear operators. On the basis of the above research, in this paper, we first provide a sufficient condition for an unbounded operator to be non-wandering operator(see Theorem 1), and apply the condition to give a necessary and sufficient condition in order that T be a non-wandering operator. (see Theorem 2)

Basic notation and definitions

Main results

Non-wandering operator in Bargmann space

Conclusion