Definition 1. For each ξ ∈ ℋ, we define the (p, q)-creation operator a^∗(ξ) and the (p, q)-annihilation operator a(ξ) on the dense subspace $$ as follows:

The (p, q)-creation and (p, q)-annihilation operators fulfill the (p, q)-commutation relations of the (p, q)-deformed quantum oscillator algebra, i.e.,

Now, we will introduce noncommutative analogs of Gaussian processes (white noise measure) for the relation of the (p, q)-deformed quantum oscillator algebra. For t ∈ ℝ, if we denote by b_t and $$ the standard pointwise annihilation and creation operators on ℱ_p,q(ℋ) defined by

Now, the (p, q)-white noise is defined by

Thus, by using (21), we deduce that ω(t) is an operator-valued distribution which satisfies

Moreover, for each $$, we define a monomial of ω by

Using the Cauchy–Schwarz inequality, we easily conclude that (24) indeed identifies a bounded linear operator in ℱ_p,q(ℋ).

Let $$ denote the complex unital ∗-algebra generated by {〈ω, ξ〉, ξ ∈ ℋ}, i.e., the algebra of noncommutative polynomials in the variables 〈ω, ξ〉. Evidently, $$ consists of all noncommutative polynomials in ω which are of the form:

In particular, elements of $$ are linear operators acting on ℱ_p,q(ℋ).

Definition 2. Let τ be a vacuum state on $$ defined by

For n ∈ ℕ\{0}, we denote by $$ the subset of $$ consisting of all noncommutative polynomials of order ≤n, i.e., all $$ given as in (25) with k ≤ n. Let $$ denote the closure of $$ in L²(τ), and let $$ be the set of orthogonal polynomials of order n defined by

Theorem 1. For each $$, define UT≔TΩ. Then, U is extended by continuity to a unitary operator U : L²(τ)⟶ℱ_p,q(ℋ) defined as follows:

Proof. Firstly from equation (27), it is clear that U : L²(τ)⟶ℱ_p,q(ℋ) is extended by continuity to a unitary operator. Moreover, since $$ is dense in L²(τ), we get the orthogonal decomposition

On the other hand, we have

Definition 3. The space of (p, q)-white noise test functions $$ is defined as a projective system of Hilbert space $$, where $$ is the set of function φ of the form

Moreover, if $$ is the set of functions Φ of the form

Theorem 2. Assume that the Young function θ satisfies the following condition:

Then, we obtain the so-called (p, q)-white noise Gel’fand triple of Hilbert spaces

Proof. Let $$. By definition, we have

On the other hand, condition (52) guarantees the existence of two constant numbers a > 0 and b > 0 such that

Then, by a simple calculus, one can see that

Hence, by using the fact that

Therefore, for γ < (2be)⁻¹, we have $$. Thus, (55) becomes

Moreover, one can see that $$ is the dual of $$ with respect to L²(τ), and we obtain the nuclear Gel’fand triple given by (53). From here the statement follows.

Now our goal is to derive a characterization of the space of (p, q)-white noise generalized functions by using a suitable space of (p, q)-entire functions with certain growth determined by using the Young functions and a suitable (p, q)-exponential map.

Let (ℬ, ‖⋅‖) be a complex Banach space. Define the space ℬ^∞ by

Then, (ℬ^∞, ‖⋅‖_∞) becomes a Banach space.

Definition 4. Let $$ be a fixed Hilbert space. A ℂ-valued function F is said to be (p, q)-entire function on $$, if there exists $$ with $$ such that

For s ∈ ℝ and γ > 0, let Γ_p,q,θ,γ(ℰ_s,ℂ) be the space of (p, q)-entire functions g on the complex Hilbert space $$ such that

Note that {Γ_p,q,θ,γ(ℰ_−s,ℂ); s ∈ ℕ, γ > 0} becomes a projective system of Banach spaces as s⟶∞ and γ ↓ 0. Then, we can define

This is called the space of (p, q)-entire functions on $$ with (θ, p, q)-exponential growth of minimal type. Similarly, {Γ_p,q,θ,γ(ℰ_s,ℂ); s ∈ ℕ, γ > 0} becomes a inductive system of Banach spaces as s⟶∞ and γ ↓ 0. $$ the space of (p, q)-entire functions on ℰ_s,ℂ with (θ, p, q)-exponential growth of finite type is defined by

Lemma 1. Let $$ be given by

Proof. Fixing s^′ > s ≥ 0 and $$ such that

By definition, the series in the right hand side of (67) converges uniformly on every bounded subset of $$. Then, for every R > 0, we have the following Cauchy’s integral formula:

Therefore, by using the fact that $$, for γ > 0, we get

Let now $$ be an orthonormal basis of $$. Then, we get

This provides the desired inequality.

Lemma 2. For each ξ ∈ ℰ_ℂ, the generating function of the noncommutative polynomials $$ defined by

Proof. Let s ≥ 0 and γ > 0, and then for any ξ ∈ ℰ_s,ℂ, we have

This proves that $$, and we obtain

On the other hand, if we choose s^′ > s such that the embedding $$ is of the Hilbert–Schmidt type and γ^′ > 0 such that

This implies that $$.

As a consequence, we can define the $$-transform of a distribution $$, at ξ ∈ ℰ_ℂ, as follows:

Moreover, by using (15) and (49), we get

Theorem 3. Assume that the sequences θ_n,p,q and $$ satisfy

Proof. Let $$, and then there exist s ≥ 0 and γ > 0 such that $$, and we have

On the other hand, by inequality (75), there exist c, γ^′ > 0, and s^′ > s such that

This proves the continuity and injectivity of the $$-transform.

Conversely, given $$, then there exist s ≥ 0 and γ > 0 such that $$ with Taylor expansion

Put $$, and then (15) and (81) yield

Using the same technics as in Lemma 1, we immediately prove that for all s^′ > s such that $$ is of the Hilbert–Schmidt type the following inequality holds:

Thus, under condition (82), we obtain

On the other hand, for all γ^′ > 0 such that $$, one can see that the series $$ converges. This proves that $$ acts surjectively and that $$ is continuous.