Commutators on Fock spaces

TL;DR

This paper develops a calculus for commutators of weighted shift operators on Fock spaces, encompassing various derivatives like fractional and Dunkl, and extends the algebraic framework beyond classical structures.

Contribution

It introduces a unified framework for analyzing commutators of weighted shift operators on Fock spaces, including Gelfond-Leontiev derivatives and other known operators.

Findings

Established a calculus for commutators on Fock spaces.

Unified treatment of fractional derivatives and Dunkl operators.

Extended algebraic structures beyond Weyl-Heisenberg algebra.

Abstract

Given a weighted $\ell^2$ space with weights associated to an entire function, we consider pairs of weighted shift operators, whose commutators are diagonal operators, when considered as operators over a general Fock space. We establish a calculus for the algebra of these commutators and apply it to the general case of Gelfond-Leontiev derivatives. This general class of operators includes many known examples, like classic fractional derivatives and Dunkl operators. This allows us to establish a general framework which goes beyond the classic Weyl-Heisenberg algebra. Concrete examples for its application are provided.