An Analytic Characterization of (p, q)-White Noise Functionals

Department of Mathematics, College of Science Al-Zulfi, Majmaah University, P.O. Box 66, Al-Majmaah 11952, Saudi Arabia Department of Mathematics, Nabeul Preparatory Institute for Engineering Studies, Carthage University, Nabeul, Tunisia Department of Mathematics, Higher School of Sciences and Technologies of Hammam-Sousse, MaPSFA Laboratory, Sousse University, Hammam Sousse 4011, Tunisia Department of Electro Mechanics, Higher Institute of Indiustrial Systems of Gabès, Gabès 6072, Tunisia

is theory is based on the quantum decomposition of the Gaussian random variable 〈ω, ξ〉 given as the sum of creation and annihilation operators which satisfies the canonical commutative commutation relation. As a generalization by replacing the classical commutative notion of independence by some other type in a noncommutative probability space, we conclude that the noncommutative white noise theory is a generalization of classical white noise theory to the description of quantum systems. In the framework of the free probability, Alpay and Salomon [9] (see also [10]) constructed a noncommutative analog of the Kondratiev space. For q ∈ (− 1, 1), Bożejko et al. introduced q-analogs of Brownian motions and Gaussian processes in [11,12], which are governed by classical independence for q � 1 and free independence for q � 0 introduced by Voiculescu et al. in [13]. 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]. More precisely, as a generalization by using the second-parameter refinement of the q-Fock space, formulated as the (p, q)-Fock space F p,q (H) which is constructed via a direct generalization of Bożejko 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. Next, 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.
Our paper is organized as follows. Section 2 is devoted to study the (p, q)-white noise functionals with special emphasis on the chaos decomposition of the noncommutative L 2 -space with respect to the vacuum expectation τ based on orthogonalization of polynomials of (p, q)-white noise. In Section 3, firstly for a fixed Young function θ with particular condition, we construct a nuclear Gel'fand triple of test and generalized functions, and we introduce the S-transform which is our main analytical tool in working with these spaces and serves to prove a characterization of white noise functionals.

Noncommutative Orthogonal Polynomials of (p, q)-White Noise
We start with the real Gel'fand triple: where S(R) is the space of rapidly decreasing functions and S ′ (R) is the dual space, i.e., the space of tempered distributions. We denote by 〈·, ·〉 the canonical bilinear form on E ′ × E and by | · | 0 the norm of H. For notational convenience, the C-bilinear form on E C × E C is denoted by the same symbol so that |ξ| 2 0 � 〈ξ, tξ〉 holds for ξ ∈ H ≔ H C (in general, the complexification of a real vector space X is denoted by X C ). In [15], Simon has proved that the space E is a nuclear Fréchet space constructed from the Hilbert space H and the Harmonic oscillator operator where for s ≥ 0, E s is the Hilbert space corresponding to the domain of A s , i.e., We define E − s to be the completion of H with respect to | · | − s � |A − s · | 0 , and hence we obtain a chain of Hilbert space E s , s ∈ R , and one can see that Let S n denote the symmetric group of all permutations on 1, n ≔ 1, . . . , n { } and I(σ) denote the number of inversions of the permutation σ ∈ S n defined by where ♯(A) denotes the cardinality of the set A. Analogously, the pair (i, j) with i < j is called a coinversion in σ if σ(i) < σ(j). e corresponding coinversion is encoded by (i, j) and contained in the set with cardinality C(σ) ≔ ♯(Cinv(σ)). Denote F 0 (H) � ⊕ n≥0 H ⊗n the full Fock space over H with the inner product 〈·, ·〉 and F fin 0 (H) the linear span of vectors of the form ξ 1 ⊗ · · · ⊗ ξ n ∈ H ⊗n , n ∈ N, where H ⊗0 � CΩ for the vacuum vector Ω � (1, 0, 0, . . .) ∈ F 0 (H). We equip F fin 0 (H) with the inner product Recall that for p and q, two real numbers such that 0 < q < p ≤ 1, the (p, q)-factorial is defined by where [n] p,q is the (p, q)-deformation of the natural number n given by Define the operator T p,q on F fin 0 (H) by a linear extension of and put Define F (n) p,q (H) as the separable Hilbert space which coincides with H ⊗n as a set and has a scalar product: Hence, the (p, q)-Fock space denoted F p,q (H) is defined by and if we denote F fin p,q (H) the linear span of vectors of the form one can see that 〈·, ·〉 p,q on F fin p,q (H) satisfies the following useful relation: For more details about the properties of the operator T p,q and the construction of the (p, q)-Fock space, see [16]. 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 F fin p,q (H) as follows: where 〈·, ·〉 denotes the inner product on H and the symbol f ⌣ i means that f i has to be deleted in the tensor product. e (p, q)-creation and (p, q)-annihilation operators fulfill the (p, q)-commutation relations of the (p, q)-deformed quantum oscillator algebra, i.e., where such that N is the standard number operator defined by and the commutator [·, ·] is defined by For more details, one can see [16]. 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 b * t the standard pointwise annihilation and creation operators on where δ t is the delta function at t and ⊗ stands for the symmetric tensor product, then one can see that the (p, q)-creation and (p, q)-annihilation operators are given as the smeared operators in terms of b t and b * t , i.e., Now, the (p, q)-white noise is defined by us, by using (21), we deduce that ω(t) is an operatorvalued distribution which satisfies Moreover, for each ξ (n) ∈ F fin p,q (H), 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 P denote the complex unital * -algebra generated by 〈ω, ξ〉, ξ ∈ H { }, i.e., the algebra of noncommutative polynomials in the variables 〈ω, ξ〉. Evidently, P consists of all noncommutative polynomials in ω which are of the form: In particular, elements of P are linear operators acting on F p,q (H).

Definition 2.
Let τ be a vacuum state on P defined by where Ω is the vacuum vector in F p,q (H). e inner product 〈·, ·〉 L 2 (τ) on P is defined by where T * 2 is the adjoint operator of T 2 in F p,q (H). Moreover, the noncommutative L 2 -space L 2 (τ) is the Hilbert space obtained as the closure of P with respect to the norm induced by the scalar product 〈·, ·〉 L 2 (τ) .

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For n ∈ N\ 0 { }, we denote by P n the subset of P consisting of all noncommutative polynomials of order ≤n, i.e., all T ∈ P given as in (25) with k ≤ n. Let P n denote the closure of P n in L 2 (τ), and let OP n be the set of orthogonal polynomials of order n defined by where ⊖ denotes the orthogonal difference in L 2 (τ).

(p, q) White Noise Gel'fand Triple and Characterization Theorem
Recall that a Young function is a continuous, convex, and increasing function such that Define a weight sequence θ n,p,q ∞ n�0 by where θ is a Young function and E p,q is the (p, q)-exponential function defined by Let θ * n,p,q be the sequence associated with the (p, q)-polar function θ * of θ, defined by For simplicity of notation, we denote where F (n) p,q (Ε s,C ) is constructed as for F (n) p,q (H) by replacing H by the space E s,C given by equation (3). Now, suppose a pair s ≥ 0 and c > 0 is given, then for φ Hence, we obtain a projective system of Hilbert spaces where Finally, we define the nuclear space F p,q,θ (E C ) by Definition 3. e space of (p, q)-white noise test functions W p,q,θ (E C ) is defined as a projective system of Hilbert space such that equipped with Hilbertian norm the space of (p, q)-white noise generalized functions is defined by Theorem 2. Assume that the Young function θ satisfies the following condition: en, we obtain the so-called (p, q)-white noise Gel'fand triple of Hilbert spaces with the C-bilinear form on W * p,q,θ (E C ) × W p,q,θ (E C ) given by where 〈·, ·〉 p,q is the canonical C-bilinear form on E ′ ⊗ p,q n C × E ⊗ p,q n C which is compatible with the inner product of H ⊗ p,q n defined by equation (12).
On the other hand, condition (52) guarantees the existence of two constant numbers a > 0 and b > 0 such that e θ(r) ≤ ae br 2 , r > 0.
Hence, by using the fact that and the inequality (57), we obtain erefore, for c < (2be) − 1 , we have C ≡ sup n≥0 θ 2 n,p,q c n < ∞. us, (55) becomes which means that F p,q,θ (E C ) ⊂ F p,q (H) and the inclusion is continuous. On the other hand, if we put where U is the isomorphism given in eorem 1, we obtain the following diagram: Moreover, one can see that W * p,q,θ,c (E − s,C ) is the dual of W * p,q,θ,c (E s,C ) with respect to L 2 (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.
Definition 4. Let K be a fixed Hilbert space. A C-valued function F is said to be (p, q)-entire function on K ∞ , if there exists (F n ) ∞ n�0 with F n ∈ K ′ ⊗ p,q n such that where the series in the right hand side of (63) converges uniformly on every bounded subset of K ∞ . For s ∈ R and c > 0, let Γ p,q,θ,c (E s,C ) be the space of (p, q)-entire functions g on the complex Hilbert space E ∞ s,C such that ‖g‖ s,p,q,θ,c ≔ sup |g(z)|E − 1 p,q θ c|z| s , z ∈ E s,C < + ∞.
Note that Γ p,q,θ,c (E − s,C ); s ∈ N, c > 0 becomes a projective system of Banach spaces as s ⟶ ∞ and c↓0. en, we can define is is called the space of (p, q)-entire functions on E C ′ with (θ, p, q)-exponential growth of minimal type. Similarly, Γ p,q,θ,c (E s,C ); s ∈ N, c > 0 becomes a inductive system of Banach spaces as s ⟶ ∞ and c↓0. g p,q,θ (E C ) the space of (p, q)-entire functions on E s,C with (θ, p, q)-exponential growth of finite type is defined by (66) Lemma 1. Let F ∈ N p,q,θ (E C ′ ) be given by 〈z 1 ⊗ p,q · · · ⊗ p,q z n , F n 〉, where F n ∈ E ⊗ p,q n C and z → � (z k ) k∈N ∈ E ′∞ C . en, for any s ≥ 0, there exists s ′ > s such that the canonical embedding i s′,s : E s′,C ⟶ E s,C is of the Hilbert-Schmidt type, and for c > 0, we get F n s,p,q ≤ ‖|F|‖ − s′,p,q,θ,c θ n,p,q c n i s′,s � � � � � � � � n HS .
Let now e → � (e k ) k∈N be an orthonormal basis of E s′,C . en, we get 6 Journal of Mathematics