Abstract
In this paper, a characterization theorem for the S -transform of infinite dimensional distributions of noncommutative white noise corresponding to the p , q -deformed quantum oscillator algebra is investigated. We derive a unitary operator U between the noncommutative L 2 -space and the p , q -Fock space which serves to give the construction of a white noise Gel’fand triple. Next, a general characterization theorem is proven for the space of p , q -Gaussian white noise distributions in terms of new spaces of p , q -entire functions with certain growth rates determined by Young functions and a suitable p , q -exponential map.
Highlights
E aim of the present paper is to introduce a proper mathematical framework of (p, q)-white noise calculus based on the noncommutative white noise corresponding to the (p, q)-deformed oscillator algebra [14]
As a generalization by using the second-parameter refinement of the q-Fock space, formulated as the (p, q)-Fock space Fp,q(H) which is constructed via a direct generalization of Bozejko and Speicher’s framework, yielding the q-Fock space when p 1, we introduce the noncommutative analogs of Gaussian processes for the relation of the (p, q)-deformed quantum oscillator algebra
We construct a white noise Gel’fand triple, and we derive the characterization of the space of generalized functions in terms of new spaces of (p, q)-entire functions with certain growth
Summary
E aim of the present paper is to introduce a proper mathematical framework of (p, q)-white noise calculus based on the noncommutative white noise corresponding to the (p, q)-deformed oscillator algebra [14]. As a generalization by using the second-parameter refinement of the q-Fock space, formulated as the (p, q)-Fock space Fp,q(H) which is constructed via a direct generalization of Bozejko and Speicher’s framework, yielding the q-Fock space when p 1, we introduce the noncommutative analogs of Gaussian processes (white noise measure) for the relation of the (p, q)-deformed quantum oscillator algebra. The (p, q)-Fock space denoted Fp,q(H) is defined by The (p, q)-white noise is defined by ω(t) ≔ bt + b∗t .
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