Generalized $q$-Fock spaces and structural identities

TL;DR

This paper introduces a family of $q$-Fock spaces that interpolate between Hardy and Fock spaces, providing new characterizations and insights into operator interactions using $q$-calculus.

Contribution

It defines generalized $q$-Fock spaces, characterizes them via classical and $q$-calculus operators, and explores intertwining operators between backward-shift and $q$-derivative.

Findings

Characterization of $q$-Fock spaces using classical and $q$-calculus operators

Identification of intertwining operators between backward-shift and $q$-Jackson derivative

Interpolation between Hardy and Fock spaces

Abstract

Using $q$-calculus we study a family of reproducing kernel Hilbert spaces which interpolate between the Hardy space and the Fock space. We give characterizations of these spaces in terms of classical operators such as integration and backward-shift operators, and their $q$-calculus counterparts. Furthermore, these new spaces allow us to study intertwining operators between classic backward-shift operators and the q-Jackson derivative.