Weighted composition operator on quaternionic Fock space

TL;DR

This paper investigates the properties of weighted composition operators on quaternionic Fock spaces, including boundedness, compactness, isometries, anti-linear operators, and conjugation, extending classical complex analysis results to quaternionic settings.

Contribution

It provides a comprehensive characterization of weighted composition operators on quaternionic Fock spaces, introducing anti-linear operators and generalizing classical complex results.

Findings

Characterized boundedness and compactness of operators

Described all isometric composition operators

Established conditions for conjugate and symmetric operators

Abstract

In this paper, we study the weighted composition operator on the Fock space $\mf$ of slice regular functions. First, we characterize the boundedness and compactness of the weighted composition operator. Subsequently, we describe all the isometric composition operators. Finally, we introduce a kind of (right)-anti-complex-linear weighted composition operator on $\mf$ and obtain some concrete forms such that this (right)-anti-linear weighted composition operator is a (right)-conjugation. Specially, we present equivalent conditions ensuring weighted composition operators which are conjugate $\mathcal{C}_{a,b,c}-$commuting or complex $\mathcal{C}_{a,b,c}-$ symmetric on $\mf$, which generalized the classical results on $\mathcal{F}^2(\mathbb{C}).$ At last part of the paper, we exhibit the closed expression for the kernel function of $\mf.$