Abstract
In this paper, a characterization theorem for the -transform of infinite dimensional distributions of noncommutative white noise corresponding to the -deformed quantum oscillator algebra is investigated. We derive a unitary operator between the noncommutative -space and the -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 -Gaussian white noise distributions in terms of new spaces of -entire functions with certain growth rates determined by Young functions and a suitable -exponential map.
1. Introduction
The white noise distribution theory originally aiming to extend Itô theory keeping contact with Lévy’s stochastic variational calculus [1] has been developed to an efficient infinite dimensional calculus with considerable applications to quantum physics, infinite dimensional harmonic analysis, infinite dimensional differential equations, quantum stochastic calculus, and mathematical finance, see, e.g., [2–8] and the references therein. This theory is based on the quantum decomposition of the Gaussian random variable 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 , Bożejko et al. introduced -analogs of Brownian motions and Gaussian processes in [11, 12], which are governed by classical independence for and free independence for introduced by Voiculescu et al. in [13].
The aim of the present paper is to introduce a proper mathematical framework of -white noise calculus based on the noncommutative white noise corresponding to the -deformed oscillator algebra [14]. More precisely, as a generalization by using the second-parameter refinement of the -Fock space, formulated as the -Fock space which is constructed via a direct generalization of Bożejko and Speicher’s framework, yielding the -Fock space when , we introduce the noncommutative analogs of Gaussian processes (white noise measure) for the relation of the -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 -entire functions with certain growth.
Our paper is organized as follows. Section 2 is devoted to study the -white noise functionals with special emphasis on the chaos decomposition of the noncommutative -space with respect to the vacuum expectation based on orthogonalization of polynomials of -white noise. In Section 3, firstly for a fixed Young function with particular condition, we construct a nuclear Gel’fand tripleof test and generalized functions, and we introduce the -transform which is our main analytical tool in working with these spaces and serves to prove a characterization of white noise functionals.
2. Noncommutative Orthogonal Polynomials of -White Noise
We start with the real Gel’fand triple:where is the space of rapidly decreasing functions and is the dual space, i.e., the space of tempered distributions. We denote by the canonical bilinear form on and by the norm of . For notational convenience, the -bilinear form on is denoted by the same symbol so that holds for (in general, the complexification of a real vector space is denoted by ). In [15], Simon has proved that the space is a nuclear Fréchet space constructed from the Hilbert space and the Harmonic oscillator operator , i.e., , where for is the Hilbert space corresponding to the domain of , i.e.,
We define to be the completion of with respect to , and hence we obtain a chain of Hilbert space , and one can see that
Let denote the symmetric group of all permutations on and denote the number of inversions of the permutation defined bywhere denotes the cardinality of the set . Analogously, the pair with is called a coinversion in if . The corresponding coinversion is encoded by and contained in the setwith cardinality . Denote the full Fock space over with the inner product and the linear span of vectors of the form , where for the vacuum vector . We equip with the inner product
Recall that for and , two real numbers such that , the -factorial is defined bywhere is the -deformation of the natural number given by
Define the operator on by a linear extension ofand put
Define as the separable Hilbert space which coincides with as a set and has a scalar product:
Hence, the -Fock space denoted is defined byand if we denote the linear span of vectors of the formone can see that on satisfies the following useful relation:
For more details about the properties of the operator and the construction of the -Fock space, see [16].
Definition 1. For each , we define the -creation operator and the -annihilation operator on the dense subspace as follows:where denotes the inner product on and the symbol means that has to be deleted in the tensor product.
The -creation and -annihilation operators fulfill the -commutation relations of the -deformed quantum oscillator algebra, i.e.,wheresuch that is the standard number operator defined byand 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 -deformed quantum oscillator algebra. For , if we denote by and the standard pointwise annihilation and creation operators on defined bywhere is the delta function at and stands for the symmetric tensor product, then one can see that the -creation and -annihilation operators are given as the smeared operators in terms of and , i.e.,Now, the -white noise is defined byThus, by using (21), we deduce that is an operator-valued distribution which satisfiesMoreover, for each , we define a monomial of byUsing the Cauchy–Schwarz inequality, we easily conclude that (24) indeed identifies a bounded linear operator in .
Let denote the complex unital -algebra generated by , i.e., the algebra of noncommutative polynomials in the variables . Evidently, consists of all noncommutative polynomials in which are of the form:In particular, elements of are linear operators acting on .
Definition 2. Let be a vacuum state on defined bywhere is the vacuum vector in . The inner product on is defined bywhere is the adjoint operator of in . Moreover, the noncommutative -space is the Hilbert space obtained as the closure of with respect to the norm induced by the scalar product .
For , we denote by the subset of consisting of all noncommutative polynomials of order , i.e., all given as in (25) with . Let denote the closure of in , and let be the set of orthogonal polynomials of order defined bywhere denotes the orthogonal difference in .
Theorem 1. For each , define . Then, is extended by continuity to a unitary operator defined as follows:where is the orthogonal projection of the monomial onto and given recursively by
Furthermore, under the action of , the operator of the left multiplication by the monomial in (denoted by ) becomes , i.e.,
Proof. Firstly from equation (27), it is clear that is extended by continuity to a unitary operator. Moreover, since is dense in , we get the orthogonal decompositionand for each , one can see that . Hence, using the fact that is the orthogonal projection in ofon , we obtainOn the other hand, we haveand this gives . Therefore, (23) and (34) yieldfrom which (30) follows.
3. () White Noise Gel’fand Triple and Characterization Theorem
Recall that a Young function is a continuous, convex, and increasing functionsuch that
Define a weight sequence bywhere is a Young function and is the -exponential function defined by
Let be the sequence associated with the -polar function of , defined by
For simplicity of notation, we denotewhere is constructed as for by replacing by the space given by equation (3). Now, suppose a pair is given, then for with , we put
Hence, we obtain a projective system of Hilbert spaceswhere
Finally, we define the nuclear space by
Definition 3. The space of -white noise test functions is defined as a projective system of Hilbert space , where is the set of function of the formsuch thatMoreover, if is the set of functions of the formequipped with Hilbertian normthe space of -white noise generalized functions is defined by
Theorem 2. Assume that the Young function satisfies the following condition:Then, we obtain the so-called -white noise Gel’fand triple of Hilbert spaceswith the -bilinear form on given bywhere is the canonical -bilinear form on which is compatible with the inner product of defined by equation (12).
Proof. Let . By definition, we haveOn the other hand, condition (52) guarantees the existence of two constant numbers and such thatThen, by a simple calculus, one can see thatHence, by using the fact thatand the inequality (57), we obtainTherefore, for , we have . Thus, (55) becomeswhich means that and the inclusion is continuous. On the other hand, if we put , where is the isomorphism given in Theorem 1, we obtain the following diagram:Moreover, one can see that is the dual of with respect to , 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 -white noise generalized functions by using a suitable space of -entire functions with certain growth determined by using the Young functions and a suitable -exponential map.
Let be a complex Banach space. Define the space byThen, becomes a Banach space.
Definition 4. Let be a fixed Hilbert space. A -valued function is said to be -entire function on , if there exists with such thatwhere the series in the right hand side of (63) converges uniformly on every bounded subset of .
For and , let be the space of -entire functions on the complex Hilbert space such thatNote that becomes a projective system of Banach spaces as and . Then, we can defineThis is called the space of -entire functions on with -exponential growth of minimal type. Similarly, becomes a inductive system of Banach spaces as and . the space of -entire functions on with -exponential growth of finite type is defined by
Lemma 1. Let be given bywhere and . Then, for any , there exists such that the canonical embedding is of the Hilbert–Schmidt type, and for , we get
Proof. Fixing and such thatBy definition, the series in the right hand side of (67) converges uniformly on every bounded subset of . Then, for every , we have the following Cauchy’s integral formula:Therefore, by using the fact that , for , we getwhich givesLet now be an orthonormal basis of . Then, we getThis provides the desired inequality.
Lemma 2. For each , the generating function of the noncommutative polynomials defined byis an element in . Moreover, for all and , there exist , and such that
Proof. Let and , and then for any , we haveThis proves that , and we obtainOn the other hand, if we choose such that the embedding is of the Hilbert–Schmidt type and such thatand by using Lemma 1 and (77), we getThis implies that .
As a consequence, we can define the -transform of a distribution , at , as follows:Moreover, by using (15) and (49), we get
Theorem 3. Assume that the sequences and satisfyfor some constant . Then, the -transform realizes a topological isomorphism from onto the space .
Proof. Let , and then there exist and such that , and we haveOn the other hand, by inequality (75), there exist , , and such thatwhich yieldsThis proves the continuity and injectivity of the -transform.
Conversely, given , then there exist and such that with Taylor expansionPut , and then (15) and (81) yieldUsing the same technics as in Lemma 1, we immediately prove that for all such that is of the Hilbert–Schmidt type the following inequality holds:Thus, under condition (82), we obtainOn the other hand, for all such that , one can see that the series converges. This proves that acts surjectively and that is continuous.
Data Availability
All data required for this paper are included within this paper.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors extend their appreciation to the Deanship of Scientific Research at Majmaah University for funding this work under project number (RGP-2019-1).