Supercyclicity and resolvent condition for weighted composition operators

TL;DR

This paper investigates the dynamical properties of weighted composition operators on Fock spaces, establishing their non-supercyclicity and characterizing when they satisfy Ritt's resolvent growth condition, notably linking it to compactness.

Contribution

It proves that no weighted composition operator on Fock spaces is supercyclic and characterizes when these operators meet Ritt's resolvent growth condition, especially relating it to compactness.

Findings

No weighted composition operators on Fock spaces are supercyclic.

Operators satisfy Ritt's resolvent growth condition if and only if they are compact.

Non-trivial composition operators satisfy the growth condition precisely when they are compact.

Abstract

For pairs of holomorphic maps $(u,\psi)$ on the complex plane, we study some dynamical properties of the weighted composition operator $W_{(u,\psi)}$ on the Fock spaces. We prove that no weighted composition operator on the Fock spaces is supercyclic. Conditions under which the operators satisfy the Ritt's resolvent growth condition are also identified. In particular, we show that a non-trivial composition operator on the Fock spaces satisfies such a growth condition if and only if it is compact.