Cyclic Composition operators on Segal-Bargmann space-Reference-Cited by-同舟云学术

Cyclic Composition operators on Segal-Bargmann space

Published:2022-01-01 Issue:1 Volume:9 Page:127-138
ISSN:2299-3282
Container-title:Concrete Operators
language:en
Short-container-title:

Author:

Ramesh G.¹,Ranjan B. Sudip¹,Naidu D. Venku²

Affiliation:

1. Department of Mathematics , Indian Institute of Technology - Hyderabad , Kandi, Sangareddy , Telangana , India

2. Department of Mathematics , Indian Institute of Technology - Hyderabad, Kandi, Sangareddy , Telangana , India

Abstract

Abstract We study the cyclic, supercyclic and hypercyclic properties of a composition operator C ϕ on the Segal-Bargmann space ℋ(ℰ), where ϕ(z) = Az + b, A is a bounded linear operator on ℰ, b ∈ ℰ with ||A|| ⩽ 1 and A*b belongs to the range of (I – A * A)½. Specifically, under some conditions on the symbol ϕ we show that if C ϕ is cyclic then A* is cyclic but the converse need not be true. We also show that if C ϕ * is cyclic then A is cyclic. Further we show that there is no supercyclic composition operator on the space ℋ(ℰ) for certain class of symbols ϕ.

Publisher

Walter de Gruyter GmbH

Subject

Applied Mathematics,Analysis

Link

https://www.degruyter.com/document/doi/10.1515/conop-2022-0133/pdf

Reference14 articles.

1. [1] J. Agler and J. E. McCarthy, Pick interpolation and Hilbert function spaces, Graduate Studies in Mathematics, 44, American Mathematical Society, Providence, RI, 2002.

2. [2] S. I. Ansari and P. S. Bourdon, Some properties of cyclic operators, Acta Sci. Math. (Szeged) 63 (1997), no. 1-2, 195–207.

3. [3] F. Bayart and Matheron, Dynamics of linear operators, Cambridge Tracts in Mathematics, 179, Cambridge University Press, Cambridge, 2009.

4. [4] G.D. Birkhoff, Démonstration d’un théoréme élémentaire sur les fonctions entiéres. C. R. Acad. Sci. Paris, 189 (1929), 473-475.

5. [5] P. S. Bourdon and J. H. Shapiro, Cyclic phenomena for composition operators, Mem. Amer. Math. Soc. 125 (1997), no. 596, x+105 pp.