Convex-cyclic weighted composition operators and their adjoints

TL;DR

This paper characterizes convex-cyclic weighted composition operators and their adjoints on the Fock space, linking their properties to derivative powers of the symbol function and eigenvalue locations, and explores their invariant sets.

Contribution

It provides a complete characterization of convex-cyclic weighted composition operators and their adjoints on the Fock space, including conditions for invariant convex sets and eigenvalues.

Findings

Characterization of convex-cyclic weighted composition operators on the Fock space.

Identification of operators with all invariant convex sets as invariant subspaces.

Proof that no supercyclic weighted composition operators exist on the Fock space.

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 \cite{CC}.