# An Analytic Characterization ofp,q-White Noise Functionals

Authors: Anis Riahi; Amine Ettaieb; Wathek Chammam; Ziyad Ali Alhussain
Journal: Journal of Mathematics
 (2020)

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## Abstract

In this paper, a characterization theorem for theS-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 L2-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.


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## Body

## 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 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].The aim of the present paper is to introduce a proper mathematical framework ofp,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 ℱp,qℋ 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. Section2 is devoted to study the p,q-white noise functionals with special emphasis on the chaos decomposition of the noncommutative L2-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(1)Wp,q,θℰℂ⊂L2τ⊂Wp,q,θ∗ℰℂ,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.


## 2. Noncommutative Orthogonal Polynomials ofp,q-White Noise

We start with the real Gel’fand triple:(2)ℰ≔Sℝ⊂H=L2ℝ,dt⊂ℰ′≔S′ℝ,where Sℝ 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 ℰℂ×ℰℂ is denoted by the same symbol so that ξ02=ξ¯,ξ holds for ξ∈ℋ≔Hℂ (in general, the complexification of a real vector space X is denoted by Xℂ). 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 A=1+t−d2/dt2, i.e., ℰ=projlims⟶∞ℰs=∩s≥0ℰs, where for s≥0,ℰs is the Hilbert space corresponding to the domain of As, i.e.,(3)ℰs=ξ∈H;ξs=Asξ0<∞.We defineℰ−s to be the completion of H with respect to ⋅−s=A−s⋅0, and hence we obtain a chain of Hilbert space ℰs,s∈ℝ, and one can see that(4)ℰ′=indlims⟶∞ℰ−s=∪s≥0ℰ−s.LetSn denote the symmetric group of all permutations on 1,n≔1,…,n and Iσ denote the number of inversions of the permutation σ∈Sn defined by(5)Iσ=♯Invσ≔♯i,j|1≤i<j≤n,σi>σj,where ♯A denotes the cardinality of the set A. Analogously, the pair i,j with i<j is called a coinversion in σ if σi<σj. The corresponding coinversion is encoded by i,j and contained in the set(6)Cinvσ≔i,j|1≤i<j≤n,σi<σj,with cardinality Cσ≔♯Cinvσ. Denote ℱ0ℋ=⊕n≥0ℋ⊗n the full Fock space over ℋ with the inner product ⋅,⋅ and ℱ0finℋ the linear span of vectors of the form ξ1⊗⋯⊗ξn∈ℋ⊗n,n∈ℕ, where ℋ⊗0=ℂΩ for the vacuum vector Ω=1,0,0,…∈ℱ0ℋ. We equip ℱ0finℋ with the inner product(7)ξ1⊗⋯⊗ξn,η1⊗⋯⊗ηm=δn,m∏k=1nξk,ηk.Recall that forp and q, two real numbers such that 0<q<p≤1, the p,q-factorial is defined by(8)np,q!=∏k=1nkp,q,with0p,q!=1,where np,q is the p,q-deformation of the natural number n given by(9)np,q=∑i=1nqi−1pn−i=pn−qnp−q,0p,q=0.Define the operatorTp,q on ℱ0finℋ by a linear extension of(10)Tp,qΩ=Ω,Tp,qξ1⊗⋯⊗ξn=∑σ∈SnqIσpCσξσ1⊗⋯⊗ξσn,and put(11)ξ1⊗p,q⋯⊗p,qξn≔Tp,qξ1⊗⋯⊗ξn,ξi∈ℋ,i∈1,n.Defineℱp,qnℋ as the separable Hilbert space which coincides with ℋ⊗n as a set and has a scalar product:(12)fn,gnp,q≔fn,gnℱp,qnℋ=Tp,qfn,gn.Hence, thep,q-Fock space denoted ℱp,qℋ is defined by(13)ℱp,qℋ=⊕n=0+∞ℱp,qnℋ,and if we denote ℱp,qfinℋ the linear span of vectors of the form(14)ξ1⊗p,q⋯⊗p,qξn∈ℱp,qnℋ,n∈ℕ,one can see that ⋅,⋅p,q on ℱp,qfinℋ satisfies the following useful relation:(15)fn,ξ1⊗p,q⋯⊗p,qξmp,q=δn,mnp,q!fn,ξ1⊗⋯⊗ξm.For more details about the properties of the operatorTp,q and the construction of the p,q-Fock space, see [16].Definition 1.
For eachξ∈ℋ, we define the p,q-creation operator a∗ξ and the p,q-annihilation operator aξ on the dense subspace ℱp,qfinℋ as follows:(16)a∗ξΩ=ξ,a∗ξf1⊗p,q⋯⊗p,qfn=ξ⊗p,qf1⊗p,q⋯⊗p,qfn,aξΩ=0,aξf1⊗p,q⋯⊗p,qfn=∑i=1nqi−1pn−iξ,fif1⊗p,q⋯⊗p,qf⌣i⊗p,q⋯⊗p,qfn,where ⋅,⋅ denotes the inner product on ℋ and the symbol f⌣i means that fi has to be deleted in the tensor product.
Thep,q-creation and p,q-annihilation operators fulfill the p,q-commutation relations of the p,q-deformed quantum oscillator algebra, i.e.,(17)aξa∗η−pa∗ηaξ=qNξ,η1,aξa∗η−qa∗ηaξ=pNξ,η1,∀ξ,η∈ℋ,N,a∗ξ=a∗ξ,N,aξ=−aξ,where(18)qNf1⊗p,q⋯⊗p,qfn=qnf1⊗p,q⋯⊗p,qfn,such that N is the standard number operator defined by(19)Nf1⊗p,q⋯⊗p,qfn=nf1⊗p,q⋯⊗p,qfn,and the commutator ⋅,⋅ is defined by B,C=BC−CB. For more details, one can see [16].
Now, we will introduce noncommutative analogs of Gaussian processes (white noise measure) for the relation of thep,q-deformed quantum oscillator algebra. For t∈ℝ, if we denote by bt and bt∗ the standard pointwise annihilation and creation operators on ℱp,qℋ defined by(20)btfnt1,…,tn−1=∑i=1nqi−1pn−ifnt1,…,ti−1,t,ti,…tn−1,bt∗fn=δt⊗^fn,where δt 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 bt and bt∗, i.e.,(21)aξ=∫ℝξtbtdt,a∗ξ=∫ℝξtbt∗dt,∀ξ∈ℋ.
Now, thep,q-white noise is defined by(22)ωt≔bt+bt∗.
Thus, by using (21), we deduce that ωt is an operator-valued distribution which satisfies(23)ω,ξ≔∫ℝωtξtdt=a∗ξ+aξ.
Moreover, for eachξn∈ℱp,qfinℋ, we define a monomial of ω by(24)ω⊗n,ξn=∫ℝnξnt1,…,tnωt1,…,ωtndt1,…,dtn=∫ℝnξnt1,…,tnbt1∗+bt1,…,btn∗+btndt1,…,dtn.
Using the Cauchy–Schwarz inequality, we easily conclude that (24) indeed identifies a bounded linear operator in ℱp,qℋ.
LetP 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:(25)T=ξ0+∑i=1kω⊗i,ξi,ξ0∈ℂ,ξi∈ℱp,qiℋ.
In particular, elements ofP are linear operators acting on ℱp,qℋ.Definition 2.
Letτ be a vacuum state on P defined by(26)τT=TΩ,Ωℱp,qℋ,T∈P,where Ω is the vacuum vector in ℱp,qℋ. The inner product ⋅,⋅L2τ on P is defined by(27)T1,T2L2τ≔τT2∗T1=T1Ω,T2Ωℱp,qℋ,T1,T2∈P,where T2∗ is the adjoint operator of T2 in ℱp,qℋ. Moreover, the noncommutative L2-space L2τ is the Hilbert space obtained as the closure of P with respect to the norm induced by the scalar product ⋅,⋅L2τ.
Forn∈ℕ\0, we denote by Pn 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 Pn in L2τ, and let OPn be the set of orthogonal polynomials of order n defined by(28)OPn=P˜n⊖P˜n−1,where ⊖ denotes the orthogonal difference in L2τ.Theorem 1.
For eachT∈P, define UT≔TΩ. Then, U is extended by continuity to a unitary operator U:L2τ⟶ℱp,qℋ defined as follows:(29)U:ω⊗n:ξn=ξn,ξn∈ℱp,qfinℋ,where :ω⊗n:ξn is the orthogonal projection of the monomial ω⊗n,ξn onto OPn and given recursively by(30):ω:ξ=ω,ξ,ξ∈ℋ,:ω⊗n+1:ξ⊗p,qn+1=ω,ξ:ω⊗n:ξ⊗p,qn−np,qξ,ξ:ω⊗n−1:ξ⊗p,qn−1.Furthermore, under the action ofU, the operator of the left multiplication by the monomial ω,ξ in L2τ (denoted by Mω,ξ) becomes ω,ξ, i.e.,(31)UMω,ξU−1=ω,ξ.Proof.
Firstly from equation (27), it is clear that U:L2τ⟶ℱp,qℋ is extended by continuity to a unitary operator. Moreover, since P is dense in L2τ, we get the orthogonal decomposition(32)L2τ=⊕n=0+∞OPn,and for each n, one can see that UOPn=ℱp,qnℋ. Hence, using the fact that U:ω⊗n:ξn is the orthogonal projection in ℱp,qℋ of(33)Uω⊗n,ξn=ω⊗n,ξnΩ=ξn,on ℱp,qnℋ, we obtain(34)U:ω⊗n:ξn=ξn.
On the other hand, we have(35)U:ω:ξ=ξ=a∗ξ+aξΩ=ω,ξΩ=Uω,ξ,∀ξ∈ℋ,and this gives :ω:ξ=ω,ξ. Therefore, (23) and (34) yield(36)Uω,ξ:ω⊗n:ξ⊗p,qn=ω,ξξ⊗p,qn=a∗ξ+aξξ⊗p,qn=ξ⊗p,qn+1+np,qξ,ξξ⊗p,qn−1=U:ω⊗n+1:ξ⊗p,qn+1+np,qξ,ξ:ω⊗n−1:ξ⊗p,qn−1,from which (30) follows.


## 3. (p,q) White Noise Gel’fand Triple and Characterization Theorem

Recall that a Young function is a continuous, convex, and increasing function(37)θ:0,+∞⟶0,+∞,such that(38)θ0=0,limx⟶+∞θxx=+∞.Define a weight sequenceθn,p,qn=0∞ by(39)θn,p,q=infx>0Ep,qθxxn,n=0,1,2,…,where θ is a Young function and Ep,q is the p,q-exponential function defined by(40)Ep,qx=∑n=0+∞qnn−1/2np,q!xn,x∈ℝ.Letθn,p,q∗ be the sequence associated with the p,q-polar function θ∗ of θ, defined by(41)θ∗x=Ep,q−1supt≥0Ep,qxtEp,qθt,x≥0.For simplicity of notation, we denote(42)φns,p,q=φnℱp,qnΕs,ℂ,s∈ℤ,where ℱp,qnΕs,ℂ is constructed as for ℱp,qnℋ by replacing ℋ by the space ℰs,ℂ given by equation (3). Now, suppose a pair s≥0andγ>0 is given, then for φ⟶=φnn=0∞ with φn∈ℱp,qnℰs,ℂ, we put(43)φ⟶s,p,q,θ,γ2≔∑n=0+∞θn,p,q−2γ−nφns,p,q2.Hence, we obtain a projective system of Hilbert spaces(44)ℱp,qℰs,ℂ,γ,θ;s≥0,γ>0,where(45)ℱp,qℰs,ℂ,γ,θ=φ⟶=φnn=0∞;φn∈ℱp,qnℰs,ℂ,φ⟶s,p,q,θ,γ<∞.Finally, we define the nuclear spaceℱp,q,θℰℂ by(46)ℱp,q,θℰℂ=projlims⟶+∞;γ↓0ℱp,qℰs,ℂ,γ,θ.Definition 3.
The space ofp,q-white noise test functions Wp,q,θℰℂ is defined as a projective system of Hilbert space Wp,q,θ,γℰs,ℂ;s≥0,γ>0, where Wp,q,θ,γℰs,ℂ is the set of function φ of the form(47)φω=∑n=0+∞:ω⊗n:φn,φ⟶=φn∈ℱp,q,θℰℂ,such that(48)φs,p,q,θ,γ≡φ⟶s,p,q,θ,γ<∞.
Moreover, ifWp,q,θ,γℰ−s,ℂ is the set of functions Φ of the form(49)Φω=∑n=0+∞:ω⊗n:Φn,Φ⟶=Φn∈ℱp,q,θ∗ℰℂ,equipped with Hilbertian norm(50)Φ−s,p,q,θ,γ2≔∑n=0+∞θn,p,q2γnΦn−s,p,q2,the space of p,q-white noise generalized functions is defined by(51)Wp,q,θ∗ℰℂ≡∪s≥0,γ>0Wp,q,θ,γℰ−s,ℂ.Theorem 2.
Assume that the Young functionθ satisfies the following condition:(52)limsupx⟶∞θxx2<∞.
Then, we obtain the so-calledp,q-white noise Gel’fand triple of Hilbert spaces(53)Wp,q,θℰℂ⟶L2τ⟶Wp,q,θ∗ℰℂ,with the ℂ-bilinear form on Wp,q,θ∗ℰℂ×Wp,q,θℰℂ given by(54)Φ,φ=∑n=0+∞Φn,φnp,q,where ⋅,⋅p,q is the canonical ℂ-bilinear form on ℰℂ′⊗p,qn×ℰℂ⊗p,qn which is compatible with the inner product of ℋ⊗p,qn defined by equation (12).Proof.
Letφ⟶=φnn=0∞∈ℱp,q,θℰℂ. By definition, we have(55)φ⟶ℱp,qℋ2=∑n=0∞φn,φnp,q=∑n=0∞θn,p,q2γnθn,p,q−2γ−nφnp,q2.
On the other hand, condition (52) guarantees the existence of two constant numbers a>0 and b>0 such that(56)eθr≤aebr2,r>0.
Then, by a simple calculus, one can see that(57)infr>0eθrrn≤a2benn/2.
Hence, by using the fact that(58)Ep,qr=∑k=0∞qkk−1/2rkk!k!kp,q!≤∑k=0∞qkk−1k!kp,q2!1/2er2/2,r>0,and the inequality (57), we obtain(59)θn,p,q2γn≤a22beγnn∑k=0∞qkk−1k!kp,q2!.
Therefore, forγ<2be−1, we have C≡supn≥0θn,p,q2γn<∞. Thus, (55) becomes(60)φ⟶ℱp,qℋ2≤C∑n=0∞θn,p,q−2γ−nφnℱp,qnℋ2=Cφ⟶0,p,q,θ,γ2≤Cφ⟶s,p,q,θ,γ2,which means that ℱp,q,θℰℂ⊂ℱp,qℋ and the inclusion is continuous. On the other hand, if we put Wp,q,θℰℂ≔U−1ℱp,q,θℰℂ, where U is the isomorphism given in Theorem 1, we obtain the following diagram:(61)ℱp,q,θℰℂ⟶ℱp,qℋ↓↓Wp,q,θℰℂ⟶L2τ.
Moreover, one can see thatWp,q,θ,γ∗ℰ−s,ℂ is the dual of Wp,q,θ,γ∗ℰs,ℂ with respect to L2τ, 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 ofp,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.
Letℬ,⋅ be a complex Banach space. Define the space ℬ∞ by(62)ℬ∞≔ξ⟶=ξkk∈ℕ;ξk∈ℬ,ξ⟶∞≔supk∈ℕξk<∞.
Then,ℬ∞,⋅∞ becomes a Banach space.Definition 4.
LetK be a fixed Hilbert space. A ℂ-valued function F is said to be p,q-entire function on K∞, if there exists Fnn=0∞ with Fn∈K′⊗p,qn such that(63)Fξ⟶=∑n=0∞Fn,ξ1⊗p,q⋯⊗p,qξn,ξ⟶=ξkk∈ℕ∈K∞,where the series in the right hand side of (63) converges uniformly on every bounded subset of K∞.
Fors∈ℝ and γ>0, let Γp,q,θ,γℰs,ℂ be the space of p,q-entire functions g on the complex Hilbert space ℰs,ℂ∞ such that(64)gs,p,q,θ,γ≔supgzEp,q−1θγzs,z∈ℰs,ℂ<+∞.
Note thatΓp,q,θ,γℰ−s,ℂ;s∈ℕ,γ>0 becomes a projective system of Banach spaces as s⟶∞ and γ↓0. Then, we can define(65)Np,q,θℰℂ′≔projlims⟶∞;γ↓0Γp,q,θ,γℰ−s,ℂ.
This is called the space ofp,q-entire functions on ℰℂ′ with θ,p,q-exponential growth of minimal type. Similarly, Γp,q,θ,γℰs,ℂ;s∈ℕ,γ>0 becomes a inductive system of Banach spaces as s⟶∞ and γ↓0. gp,q,θℰℂ the space of p,q-entire functions on ℰs,ℂ with θ,p,q-exponential growth of finite type is defined by(66)gp,q,θℰℂ=∪s≥0,γ>0Γp,q,θ,γℰs,ℂ.Lemma 1.
LetF∈Np,q,θℰℂ′ be given by(67)Fz⟶=∑n=0∞z1⊗p,q⋯⊗p,qzn,Fn,where Fn∈ℰℂ⊗p,qn and z⟶=zkk∈ℕ∈ℰℂ′∞. Then, for any s≥0, there exists s′>s such that the canonical embedding is′,s:ℰs′,ℂ⟶ℰs,ℂ is of the Hilbert–Schmidt type, and for γ>0, we get(68)Fns,p,q≤F−s′,p,q,θ,γθn,p,qγnis′,sHSn.Proof.
Fixings′>s≥0 and z⟶=zkk∈ℕ∈ℰ−s′,ℂ∞ such that(69)zk−s′=1,∀k∈ℕ.
By definition, the series in the right hand side of (67) converges uniformly on every bounded subset of ℰ−s′,ℂ∞. Then, for every R>0, we have the following Cauchy’s integral formula:(70)z1⊗p,q⋯⊗p,qzn,Fn=12iπ∫λ=RFλz⟶λn+1dλ,n∈ℕ.
Therefore, by using the fact thatF∈Np,q,θℰℂ′, for γ>0, we get(71)z1⊗p,q⋯⊗p,qzn,Fn≤F−s′,p,q,θ,γEp,qθγRγRnγn,∀R>0,which gives(72)z1⊗p,q⋯⊗p,qzn,Fn≤F−s′,p,q,θ,γθn,p,qγn.
Let nowe⟶=ekk∈ℕ be an orthonormal basis of ℰs′,ℂ. Then, we get(73)Fns,p,q2≤∑k1,…,kn≥0Fn,ek1⊗p,q⋯⊗p,qekn2≤∑k1,…,kn≥0F−s′,p,q,θ,γ2θn,p,q2γ2n∏j=1nekjs′2≤F−s′,p,q,θ,γ2θn,p,q2γ2nis′,s⊗nHS2≤F−s′,p,q,θ,γ2θn,p,q2γ2nis′,sHS2n.
This provides the desired inequality.Lemma 2.
For eachξ∈ℰℂ, the generating function of the noncommutative polynomials :ω⊗n:ξ⊗p,qn defined by(74):Ep,qω,ξ≔∑n=0∞qnn−1/2np,q!:ω⊗n:ξ⊗p,qn,is an element in Wp,q,θℰℂ. Moreover, for all s≥0 and γ>0, there exist c,γ′>0, and s′>s such that(75):Ep,qω,ξ:s,p,q,θ,γ≤cEp,qθ∗ξs′γ′.Proof.
Lets≥0 and γ>0, and then for any ξ∈ℰs,ℂ, we have(76)Ep,qz,ξEp,q−1θγz−s≤Ep,qz−sξsEp,q−1θγz−s≤Ep,qθ∗ξsγ.
This proves thatEp,q⋅,ξ∈Np,q,θℰℂ′, and we obtain(77)Ep,q⋅,ξ−s,p,q,θ,γ=supz∈ℰ−sEp,qz,ξEp,q−1θγz−s≤Ep,qθ∗ξsγ.
On the other hand, if we chooses′>s such that the embedding is′,s:ℰs′,ℂ⟶ℰs,ℂ is of the Hilbert–Schmidt type and γ′>0 such that(78)γ′γis′,sHS<1,and by using Lemma 1 and (77), we get(79):Ep,qω,ξ:s,p,q,θ,γ2=∑n=0∞θn,p,q−2γ−nqnn−1np,q2!ξ⊗p,qns,p,q2≤Ep,q⋅,ξ−s′,p,q,θ,γ′2∑n=0∞qnn−1γ′γis′,sHS2n≤c2Ep,q2θ∗ξs′γ′.
This implies thatEp,qω,ξ:∈Wp,q,θℰℂ.
As a consequence, we can define theS-transform of a distribution Φ∈Wp,q,θ∗ℰℂ, at ξ∈ℰℂ, as follows:(80)SΦξ≔Φ,Ep,q⋅,ξ:.
Moreover, by using (15) and (49), we get(81)SΦξ=∑n=0+∞qnn−1/2np,q!Φn,ξ⊗p,qnp,q=∑n=0+∞qnn−1/2Φn,ξ⊗n.Theorem 3.
Assume that the sequencesθn,p,q and θn,p,q∗ satisfy(82)θn,p,qθn,p,q∗≤βnqnn−1/2np,q!,for some constant β>0. Then, the S-transform realizes a topological isomorphism from Wp,q,θ∗ℰℂ onto the space gp,q,θ∗ℰℂ.Proof.
LetΦ∈Wp,q,θ∗ℰℂ, and then there exist s≥0 and γ>0 such that Φ∈Wp,q,θ,γℰ−s,ℂ, and we have(83)SΦξ≤∑n=0+∞θn,p,qγn/2Φn−s,p,qθn,p,q−1γ−n/2qnn−1/2np,q!ξ⊗ns≤∑n=0+∞θn,p,q2γnΦn−s,p,q21/2∑n=0+∞qnn−1θn,p,q−2γnnp,q!2ξs2n1/2.
On the other hand, by inequality (75), there exist c, γ′>0, and s′>s such that(84)SΦξ≤cΦ−s,p,q,θ,γEp,qθ∗ξs′γ′,which yields(85)SΦs′,p,q,θ∗,γ′−1≤cΦ−s,p,q,θ,γ.
This proves the continuity and injectivity of theS-transform.
Conversely, giveng∈gp,q,θ∗ℰℂ, then there exist s≥0 and γ>0 such that g∈gp,q,θ∗,γℰs,ℂ with Taylor expansion(86)gξ=∑n=0+∞Gn,ξ⊗p,qn,Gn∈ℰℂ′⊗p,qn.
PutΦω=∑n=0+∞qn1−n/2np,q!:ω⊗n:Gn, and then (15) and (81) yield(87)SΦξ=∑n=0+∞np,q!Gn,ξ⊗n=∑n=0+∞Gn,ξ⊗p,qn=gξ.
Using the same technics as in Lemma1, we immediately prove that for all s′>s such that is′,s is of the Hilbert–Schmidt type the following inequality holds:(88)Gn−s′,p,q≤gs,p,q,θ∗,γθn,p,q∗γnis′,sHSn.
Thus, under condition (82), we obtain(89)Φ−s′,p,q,θ,γ′2=∑n=0+∞qn1−nnp,q2!θn,p,q2γ′nGn−s′,p,q2≤∑n=0+∞qn1−nnp,q2!θn,p,q2γ′nθn,p,q∗2γ2ngs,p,q,θ∗,γ2is′,sHS2n≤gs,p,q,θ∗,γ2∑n=0+∞βγγ′is′,sHS2n.
On the other hand, for allγ′>0 such that βγγ′is′,sHS<1, one can see that the series ∑n=0+∞βγγ′is′,sHS2n converges. This proves that S acts surjectively and that S−1 is continuous.



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*Source: 6319138-2020-12-08.xml*