Spectrums and uniform mean ergodicity of weighted composition operators on Fock spaces

TL;DR

This paper investigates the spectral properties and ergodic behavior of weighted composition operators on Fock spaces, providing conditions for power boundedness and uniform mean ergodicity based on symbol values.

Contribution

It introduces new criteria for power boundedness and uniform mean ergodicity of weighted composition operators on Fock spaces, linking these properties to symbol evaluations.

Findings

Operators with specific symbol values are power bounded.

Conditions for uniform mean ergodicity are established.

Spectral analysis supports ergodic property results.

Abstract

For holomorphic pairs of symbols $(u, \psi)$, we study various structures of the weighted composition operator $ W_{(u,\psi)} f= u \cdot f(\psi)$ defined on the Fock spaces $\mathcal{F}_p$. We have identified operators $W_{(u,\psi)}$ that have power bounded and uniformly mean ergodic properties on the spaces. These properties are described in terms of easy to apply conditions relying on the values $|u(0)|$ and $|u(\frac{b}{1-a})|$ where $ a$ and $b$ are coefficients from linear expansion of the symbol $\psi$. The spectrum of the operators are also determined and applied further to prove results about uniform mean ergodicity.