Concrete Operators (Nov 2022)

Cyclic Composition operators on Segal-Bargmann space

  • Ramesh G.,
  • Ranjan B. Sudip,
  • Naidu D. Venku

DOI
https://doi.org/10.1515/conop-2022-0133
Journal volume & issue
Vol. 9, no. 1
pp. 127 – 138

Abstract

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We study the cyclic, supercyclic and hypercyclic properties of a composition operator Cϕ 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 Cϕ is cyclic then A* is cyclic but the converse need not be true. We also show that if Cϕ* 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.

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