Convex-Cyclic Weighted Composition Operators and Their Adjoints

Tesfa Mengestie

doi:10.1007/s00009-021-01812-7

Abstract

We characterize the convex-cyclic weighted composition operators W_{(u,psi )} and their adjoints on the Fock space in terms of the derivative powers of psi and the location of the eigenvalues of the operators on the complex plane. Such a description is also equivalent to identifying the operators or their adjoints for which their invariant closed convex sets are all invariant subspaces. We further show that the space supports no supercyclic weighted composition operators with respect to the pointwise convergence topology and, hence, with the weak and strong topologies, and answers a question raised by T. Carrol and C. Gilmore in [5].

Highlights

The study of the weighted composition operator W(u,ψ) : f → u · f (ψ) with symbol ψ and multiplier u acting on various spaces of holomorphic functions traces back to works related to isometries on the Hardy spaces [11,14] and commutants of Toeplitz operators [8,9]
We are interested in linear dynamical properties of the operators and their adjoints on the Fock space F2 which consists of square integrable analytic functions in C with respect to the Gaussian measure dμ(z)
Having observed the absence of supercyclic weighted composition operators and their adjoints on the Fock space, we considered the cyclicity problem in [19] and proved the following interesting result

Summary

Introduction

The study of the weighted composition operator W(u,ψ) : f → u · f (ψ) with symbol ψ and multiplier u acting on various spaces of holomorphic functions traces back to works related to isometries on the Hardy spaces [11,14] and commutants of Toeplitz operators [8,9]. We are interested in linear dynamical properties of the operators and their adjoints on the Fock space F2 which consists of square integrable analytic functions in C with respect to the Gaussian measure dμ(z). A bounded linear operator T on a separable Banach space H is said to be cyclic if there exists a vector f in H for which the span of the orbit. It is known that every bounded linear operator on an infinite-dimensional complex separable Hilbert space is the sum of two hypercyclic operators [1, p. This result holds true with the summands being cyclic operators as well [24]

Cyclic Weighted Composition Operators and Their Adjoints

Convex-Cyclic Weighted Composition Operators and Their Adjoints

Invariant Convex Sets for Weighted Composition Operators and Their Adjoints

Weak and τpt -Supercyclic Weighted Composition Operators