Dynamics of the Volterra-type integral and differentiation operators on generalized Fock spaces

TL;DR

This paper investigates the dynamical behaviors of differentiation and Volterra-type integral operators on generalized Fock spaces, revealing conditions for properties like supercyclicity, hypercyclicity, and ergodicity, and establishing links with Ritt's resolvent condition.

Contribution

It provides a comprehensive analysis of the dynamical properties of these operators on generalized Fock spaces, including new characterizations and conditions for various operator behaviors.

Findings

Differentiation operator is always supercyclic on these spaces.

Characterization of when the differentiation operator is hypercyclic, power bounded, and ergodic.

Operators satisfy Ritt's resolvent condition if and only if they are power bounded and ergodic.

Abstract

Various dynamical properties of the differentiation and Volterra-type integral operators on generalized Fock spaces are studied. We show that the differentiation operator is always supercyclic on these spaces. We further characterize when it is hypercyclic, power bounded and uniformly mean ergodic. We prove that the operator satisfies the Ritt's resolvent condition if and only if it is power bounded and uniformly mean ergodic. Some similar results are obtained for the Volterra-type and Hardy integral operators.