Wick calculus for noncommutative white noise corresponding to $q$-deformed commutation relations

TL;DR

This paper develops a Wick calculus framework for noncommutative white noise associated with $q$-deformed commutation relations, establishing algebraic structures and inequalities within this setting.

Contribution

It introduces a novel Wick calculus for $q$-deformed noncommutative white noise and constructs a Gel'fand triple with algebraic properties.

Findings

Established a Gel'fand triple for $q$-deformed Fock space

Proved a V {a}ge-type inequality for the Wick product

Constructed algebraic structures for test and generalized functionals

Abstract

We derive the Wick calculus for test and generalized functionals of noncommutative white noise corresponding to $q$-deformed commutation relations with $q\in(-1,1)$. We construct a Gel'fand triple centered at the $q$-deformed Fock space in which both the test, nuclear space and its dual space are algebras with respect to the addition and the Wick multiplication. Furthermore, we prove a V\r{a}ge-type inequality for the Wick product on the dual space.