Spectral theory and Wiener-Itô decomposition for the image of a Jacobi field

Assume that K⁺ : H_ → T_ is a bounded operator, where H_ and T_ are Hilbert spaces and ρ is a measure on the space H_. Denote by ρK the image of the measure ρ under K⁺. This paper aims to study the measure ρK assuming ρ to be the spectral measure of a Jacobi field. We obtain a family of operators...

Ausführliche Beschreibung

Gespeichert in:

Bibliographische Detailangaben
Datum:	2007
Hauptverfasser:	Berezansky, Yu.M., Pulemyotov, A.D.
Format:	Artikel
Sprache:	English
Veröffentlicht:	Інститут математики НАН України 2007
Schriftenreihe:	Український математичний журнал
Schlagworte:	Статті
Online Zugang:	http://dspace.nbuv.gov.ua/handle/123456789/164192
Tags:	Tag hinzufügen Keine Tags, Fügen Sie den ersten Tag hinzu!
Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Zitieren:	Spectral theory and Wiener-Itô decomposition for the image of a Jacobi field / Yu.M. Berezansky, A.D. Pulemyotov // Український математичний журнал. — 2007. — Т. 59, № 6. — С. 744–763. — Бібліогр.: 30 назв. — англ.

Institution

Digital Library of Periodicals of National Academy of Sciences of Ukraine

id	irk-123456789-164192
record_format	dspace
spelling	irk-123456789-1641922020-02-09T01:26:43Z Spectral theory and Wiener-Itô decomposition for the image of a Jacobi field Berezansky, Yu.M. Pulemyotov, A.D. Статті Assume that K⁺ : H_ → T_ is a bounded operator, where H_ and T_ are Hilbert spaces and ρ is a measure on the space H_. Denote by ρK the image of the measure ρ under K⁺. This paper aims to study the measure ρK assuming ρ to be the spectral measure of a Jacobi field. We obtain a family of operators whose spectral measure equals ρK. We also obtain an analogue of the Wiener – Ito decomposition for ˆ ρK. Finally, we illustrate the results obtained by carrying out the explicit calculations for the case, where ρK is a Levy noise measure. Припустимо, що K⁺:H_→T_ є обмеженим оператором, де H_ та T_ – гільбертові простори, i p – міра на просторі H_. Позначимо через ρK зображення міри ρ під дією K⁺. Метою цієї роботи є вивчення міри ρK за припущення, що ρ є спектральною мірою поля Якобі. Отримано сім'ю операторів із спектральною мірою, рівною ρK, а також аналог розкладу Вінера – Іто для ρK. Одержані результати проілюстровано явними розрахунками для випадку, коли ρK є мірою шуму Леві. 2007 Article Spectral theory and Wiener-Itô decomposition for the image of a Jacobi field / Yu.M. Berezansky, A.D. Pulemyotov // Український математичний журнал. — 2007. — Т. 59, № 6. — С. 744–763. — Бібліогр.: 30 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/164192 517.9 en Український математичний журнал Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Статті Статті
spellingShingle	Статті Статті Berezansky, Yu.M. Pulemyotov, A.D. Spectral theory and Wiener-Itô decomposition for the image of a Jacobi field Український математичний журнал
description	Assume that K⁺ : H_ → T_ is a bounded operator, where H_ and T_ are Hilbert spaces and ρ is a measure on the space H_. Denote by ρK the image of the measure ρ under K⁺. This paper aims to study the measure ρK assuming ρ to be the spectral measure of a Jacobi field. We obtain a family of operators whose spectral measure equals ρK. We also obtain an analogue of the Wiener – Ito decomposition for ˆ ρK. Finally, we illustrate the results obtained by carrying out the explicit calculations for the case, where ρK is a Levy noise measure.
format	Article
author	Berezansky, Yu.M. Pulemyotov, A.D.
author_facet	Berezansky, Yu.M. Pulemyotov, A.D.
author_sort	Berezansky, Yu.M.
title	Spectral theory and Wiener-Itô decomposition for the image of a Jacobi field
title_short	Spectral theory and Wiener-Itô decomposition for the image of a Jacobi field
title_full	Spectral theory and Wiener-Itô decomposition for the image of a Jacobi field
title_fullStr	Spectral theory and Wiener-Itô decomposition for the image of a Jacobi field
title_full_unstemmed	Spectral theory and Wiener-Itô decomposition for the image of a Jacobi field
title_sort	spectral theory and wiener-itô decomposition for the image of a jacobi field
publisher	Інститут математики НАН України
publishDate	2007
topic_facet	Статті
url	http://dspace.nbuv.gov.ua/handle/123456789/164192
citation_txt	Spectral theory and Wiener-Itô decomposition for the image of a Jacobi field / Yu.M. Berezansky, A.D. Pulemyotov // Український математичний журнал. — 2007. — Т. 59, № 6. — С. 744–763. — Бібліогр.: 30 назв. — англ.
series	Український математичний журнал
work_keys_str_mv	AT berezanskyyum spectraltheoryandwieneritodecompositionfortheimageofajacobifield AT pulemyotovad spectraltheoryandwieneritodecompositionfortheimageofajacobifield
first_indexed	2025-07-14T16:42:45Z
last_indexed	2025-07-14T16:42:45Z
_version_	1837641358392688640
fulltext	UDC 517.9 Yu. M. Berezansky (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv), A. D. Pulemyotov (Cornell Univ., Ithaca, USA) THE SPECTRAL THEORY AND THE WIENER – ITÔ DECOMPOSITION FOR THE IMAGE OF A JACOBI FIELD∗ SPEKTRAL\NA TEORIQ TA ROZKLAD VINERA – ITO DLQ ZOBRAÛENNQ POLQ QKOBI Assume that K+ : H− → T− is a bounded operator, where H− and T− are Hilbert spaces and ρ is a measure on the space H−. Denote by ρK the image of the measure ρ under K+. This paper aims to study the measure ρK assuming ρ to be the spectral measure of a Jacobi field. We obtain a family of operators whose spectral measure equals ρK . We also obtain an analogue of the Wiener – Itô decomposition for ρK . Finally, we illustrate the results obtained by carrying out the explicit calculations for the case, where ρK is a Lévy noise measure. Prypustymo, wo K+ : H− → T− [ obmeΩenym operatorom, de H− ta T− — hil\bertovi prostory, i ρ — mira na prostori H−. Poznaçymo çerez ρK zobraΩennq miry ρ pid di[g K+. Metog ci[] roboty [ vyvçennq miry ρK za prypuwennq, wo ρ [ spektral\nog mirog polq Qkobi. Otrymano sim’g operatoriv iz spektral\nog mirog, rivnog ρK , a takoΩ analoh rozkladu Vinera – Ito dlq ρK . OderΩani rezul\taty proilgstrovano qvnymy rozraxunkamy dlq vypadku, koly ρK [ mirog ßumu Levi. 1. Introduction. Consider a real separable Hilbert space H and a rigging H− ⊃ H ⊃ H+ with the pairing 〈·, ·〉H . We assume the embedding H+ ↪→ H to be a Hilbert – Schmidt operator. Consider another real separable Hilbert space T and a rigging T− ⊃ T ⊃ T+ with the pairing 〈·, ·〉T . Given a bounded operator K : T+ → H+, define the operator K+ : H− → T− via the formula 〈K+ξ, f〉T = 〈ξ,Kf〉H , ξ ∈ H−, f ∈ T+. Let ρ be a Borel probability measure on the space H−. We denote by ρK the image of the measure ρ under the mapping K+. This paper aims to study the measure ρK assuming ρ to be the spectral measure of a Jacobi field J = (J̃(φ))φ∈H+ . In particular, we want to explain the Wiener – Itô decomposition for ρK exploiting the operator theory point of view. Our approach is based on the well-known connection between Jacobi matrices and orthogonal polynomials and the infinite-dimensional version of this connection. The principal examples will be described below. De facto, this paper develops the ideas of [22]. The assumption of the density of Ran(K) played a crucial role in [22]. This prevented the results from covering an im- portant case of a Lévy noise measure. In the present paper, we show how to develop the theory without assuming the density of Ran(K). The construction here is much more general, and it allows to study more complicated phenomena. In particular, it describes the Lévy noise measure as detailed below. By definition, a Jacobi field J = (J̃(φ))φ∈H+ is a family of commuting selfadjoint three-diagonal operators J̃(φ) acting in the Fock space ∗ The research was partially supported by DFG (Project 436 UKR 113/78/0-1) and by INTAS (Project 00- 257). c© Yu. M. BEREZANSKY, A. D. PULEMYOTOV, 2007 744 ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6 THE SPECTRAL THEORY AND THE WIENER – ITÔ DECOMPOSITION FOR THE IMAGE ... 745 F(H) = ∞⊕ n=0 Fn(H), Fn(H) = H⊗̂n C (we suppose H⊗̂0 C = C). The operators J̃(φ) are assumed to depend on the indexing parameter φ ∈ H+ linearly and continuously. In Section 2 of the present paper, we adduce the rigorous definition and the basic spectral theory of a Jacobi field. More details can be found in, e.g., [1 – 4]. Remark that the concept of a Jacobi field is relatively new, therefore the definitions given in different papers may differ in minor details. Jacobi fields are actively used in non-Gaussian white noise analysis and theory of stochastic processes, see [2, 4 – 13]. In the case of a finite-dimensional H, the theory of Jacobi fields is closely related to some results in [14 – 17]. The most principal examples of spectral measures of Jacobi fields are the the Gaussian measure and the Poisson measure. The Jacobi field with the Gaussian spectral measure is the classical free field in quantum field theory, see, e.g., [2, 3, 5, 18]. The Jacobi field with the Poisson spectral measure is the so-called Poisson field (de facto, it has been independently discovered in [19, 20]). Section 4 of the present paper contains the rigorous definition of the Poisson field. More details can be found in, e.g., [2, 3, 5, 6]. For other examples of spectral measures of Jacobi fields, see [2, 4]. In Section 3 of the present paper, for a given operator K and a given Jacobi field J, we construct a Fock-type space Fext(T+,K) = ∞⊕ n=0 Fext n (T+,K) and a family JK = (J̃K(f))f∈T+ of operators in Fext(T+,K) pursuing the three follow- ing goals: 1. To show that ρK is the spectral measure of the family JK . 2. To show that the Fourier transform corresponding to the generalized joint eigen- vector expansion of JK coincides with the generalized Wiener – Itô – Segal transform as- sociated with ρK . 3. To obtain an analogue of the Wiener – Itô orthogonal decomposition for ρK em- ploying the generalized Wiener – Itô – Segal transform associated with ρK . The family JK can no longer appear as a Jacobi field. Basically, in order to introduce it and reach our destination, we need to extend the concept of a Jacobi field. The relation between JK and the original J will become clear from the definitions given below. Several versions of the third goal for concrete examples of the measure ρK were considered over the last decade. Different approaches were utilized. A collection of ref- erences will be given later. To a large extent, this paper is intended to give an operator theory approach to the problem within a rather general setting. We expect that our con- struction will be useful in explaining the complicated technical results obtained by other methods. It is also hopeful that it will be used for generalizations and extensions of the concrete classical theories. Of course, our construction allows to deal with a wider class of measures than those investigated in this context before. We will now recall the classical concepts of the Wiener – Itô – Segal transform and the Wiener – Itô decomposition briefly. More details can be found in, e.g., [18] or [21]. ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6 746 Yu. M. BEREZANSKY, A. D. PULEMYOTOV Let γ stand for the Gaussian measure on H−. Let Pn stand for the set of all continuous polynomials on H− with their degree less than or equal to n. Denote by P̃n the closure of Pn in L2(H−, dγ). The Wiener – Itô – Segal transform Fn(H+) Φn → IΦn = 1√ n! PrP̃n�P̃n−1 〈·⊗n,Φn〉H ∈ L2(H−, dγ), n ∈ Z+ (1.1) (we suppose P̃−1 = {0} and preserve the notation 〈·, ·〉H for the pairing between Fn(H+) and Fn(H−)) can be extended to a unitary operator acting from F(H) to L2(H−, dγ). The unitarity of I implies L2(H−, dγ) = ∞⊕ n=0 ( P̃n � P̃n−1 ) = ∞⊕ n=0 I(Fn(H)). This formula constitutes the Wiener – Itô orthogonal decomposition for the Gaussian mea- sure. It is a powerful technical tool for carrying out the calculations in the space L2(H−, dγ). Remark that analogous results are possible to obtain considering the Poisson measure instead of the Gaussian measure γ. The Fourier transform corresponding to the generalized joint eigenvector expansion of the classical free field coincides with the Wiener – Itô – Segal transform I. An analogous result is possible to obtain for the Poisson field. These facts give the basis for investigating the concept of the Wiener – Itô decomposition from the viewpoint of spectral theory of Jacobi fields. The operator I can be represented as a sum of operators of multiple stochastic integra- tion. An analogous result is possible to obtain considering the Poisson measure instead of the Gaussian measure γ. Our generalization of the classical picture is as follows. Let Qn stand for the set of all continuous polynomials on T− with their degree less than or equal to n. Denote by Q̃n the closure of Qn in L2(T−, dρK). We suppose Q̃−1 = {0} and preserve the notation 〈·, ·〉T for the pairing between Fn(T+) and Fn(T−). One may introduce the generalized Wiener – Itô – Segal transform IK : Fn(T+) → L2(T−, dρK) associated with the measure ρK by the formula analogous to (1.1). Of course, now Q̃n has to appear in the place of P̃n and the pairing used must be 〈·, ·〉T . But, in general, IK cannot be extended to a unitary operator acting from F(T ) to L2(T−, dρK). We construct the space Fext(T+,K) so that IK could be extended to a unitary oper- ator acting from Fext(T+,K) to L2(T−, dρK). The orthogonal component Fext n (T+,K) has to be constructed as the completion of Fn(T+) with respect to a new scalar product (·, ·)Fext n (T+,K). Clearly, the scalar product (·, ·)Fext(T+,K) should satisfy the equality (Fn, Gn)Fext n (T+,K) = (IKFn, IKGn)L2(T−,dρK), Fn, Gn ∈ Fn(T+). Basically, the problem of constructing the space Fext(T+,K) consists in identifying the scalar product (·, ·)Fext(T+,K) explicitly. If the range Ran(K) is dense in H+ and J is the classical free field or the Poisson field, then the scalar product (·, ·)Fext(T+,K) satisfies the equality (Fn, Gn)Fext n (T+,K) = (K⊗nFn,K ⊗nGn)Fn(H), Fn, Gn ∈ Fn(T+). This case has undergone a detailed study in [22]. In the general case we are considering, the scalar product (·, ·)Fext(T+,K) has a much more complicated form. ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6 THE SPECTRAL THEORY AND THE WIENER – ITÔ DECOMPOSITION FOR THE IMAGE ... 747 In order to construct the family JK , we introduce an isometric operator A : Fext(T+, K) → F(H). The set Ran(A) is invariant with respect to the operators J̃(Kf), f ∈ T+. We define J̃K(f) via the formula J̃K(f) = A−1J̃(Kf)A, f ∈ T+. Importantly, this family is no longer a Jacobi field (at least, in general). If the range Ran(K) is dense in H+ and J is the classical free field or the Poisson field, then the operator A satisfies the equality A = ∞⊕ n=0 K⊗n (we suppose K⊗0 = IdC .) In the general case we are considering, A has a much more complicated form. The unitarity of IK implies L2(T−, dρK) = ∞⊕ n=0 ( Q̃n � Q̃n−1 ) = ∞⊕ n=0 IK(Fext n (T+,K)). This formula constitutes an analogue of the Wiener – Itô orthogonal decomposition for the measure ρK . It discovers the Fock-type structure of the space L2(T−, dρK) and enables one to carry out the calculations in L2(T−, dρK). As mentioned above, the Wiener – Itô – Segal transform I can be represented as a sum of operators of multiple stochastic integration. Presumably, an analogous representation is possible to obtain for the generalized Wiener – Itô – Segal transform IK . However, we do not concern ourselves with this problem in the present paper. Noteworthily, if K is the operator of multiplication by a function of a new indepen- dent variable and J is the Poisson field, then ρK is a Lévy noise measure on T−. This is an important illustrative example for us. Describing the operator theory approach to the Wiener – Itô decomposition for a Lévy noise measure is one of the objectives of the present work. Related questions were studied in many papers as detailed below. Our intention is to offer a construction that would help clarify the complicated phenomena oc- curring in this area. We will now discuss a few details of the definition. Namely, suppose T = L2(Rd2 , dτ) and H = L2(Rd1+d2 , d(σ ⊗ τ)). Fix a real-valued function κ on R d1 and denote by σκ the image of σ under κ. If K is defined via the formula f(t) → (Kf)(s, t) = κ(s)f(t), s ∈ R d1 , t ∈ R d2 , (1.2) and J is the Poisson field, then ρK is the Lévy noise measure on T− with the Lévy measure σκ and the intensity measure τ. (A more complicated choice of K yields the fractional Lévy noise measure on T−.) In this case, the space Fext(T+,K) should be similar to the extended Fock space investigated in [11]. A special form of this space has been introduced in [23] in the framework of Gamma white noise analysis. Its further study has been carried out in [7, 8, 10, 24], see also [12]. A family of operators with a Lévy noise spectral measure has been constructed in [9], see also [11]. The case of the Gamma measure was studied in [7]. An analogue of the Wiener – Itô decomposition for a Lévy noise measure has been obtained in [11], see also [25 – 28]. The case of the Gamma measure was studied in [8, 10, 23]. Remark that the ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6 748 Yu. M. BEREZANSKY, A. D. PULEMYOTOV works [11, 26 – 28] (respectively, [8, 10]) represent the generalized Wiener – Itô – Segal transform associated with a Lévy noise measure (respectively, the Gamma measure) as a sum of operators of stochastic integration. Once again we emphasize that the present paper does not attempt to obtain an analogous representation for the generalized Wiener – Itô – Segal transform associated with the measure ρK in the general case. While constructing the space Fext(T+,K) in Section 3, we identify the scalar product (·, ·)Fext(T+,K) in terms of the operator K and the initial Jacobi field J. However, it re- mains quite a challenge to carry out the explicit calculations for a particular operator and a particular field. In Section 4 of the present paper, we illustrate the constructions of Sec- tion 3 by carrying out the explicit calculations for the operator (1.2) and the Poisson field. Theorem 3.1 and Theorem 4.1 explain the Fock-type structure of the space L2(T−, dρK) in this case. Theorem 4.1 implies that Fext(T+,K) is similar to the extended Fock space investigated in [11]. Remark that the riggings we consider in this paper are all quasinuclear. One may consider nuclear riggings instead. 2. Commutative Jacobi fields. This section contains the definition and the basic spectral theory of a Jacobi field. Let H be a real separable Hilbert space. Denote by HC the complexification of H. Let ⊗̂ stand for the symmetric tensor product. Consider the symmetric Fock space F(H) = ∞⊕ n=0 Fn(H), Fn(H) = H⊗̂n C (we suppose H⊗̂0 C = C). This space consists of the sequences Φ = (Φn)∞n=0, Φn ∈ ∈ Fn(H). In what follows, we identify Φn ∈ Fn(H) with (0, . . . , 0,Φn, 0, 0, . . .) ∈ ∈ F(H) (Φn standing at the nth position). The finite vectors Φ = (Φ1, . . . ,Φn, 0, 0, . . .) ∈ F(H) form a linear topological space Ffin(H) ⊂ F(H). The convergence in Ffin(H) is equivalent to the uniform finite- ness and coordinatewise convergence. The vector Ω = (1, 0, 0, . . .) ∈ Ffin(H) is called vacuum. Let H− ⊃ H ⊃ H+ (2.1) be a rigging of H with real separable Hilbert spaces H+ and H− = (H+)′ (hereafter, X ′ denotes the dual of the space X). We suppose the embedding H+ ↪→ H to be a Hilbert – Schmidt operator. The pairing in (2.1) can be extended naturally to a pairing between Fn(H+) and Fn(H−). The latter can be extended to a pairing between Ffin(H+) and (Ffin(H+))′. In what follows, we use the notation 〈·, ·〉H for all of these pairings. Note that (Ffin(H+))′ coincides with the direct product of the spaces Fn(H−), n ∈ Z+. Throughout the paper, PrX F denotes the projection of a vector F onto a subspace X. 2.1. Definition of a Jacobi field. In the Fock space F(H), consider a family J = = (J (φ))φ∈H+ of operator-valued Jacobi matrices ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6 THE SPECTRAL THEORY AND THE WIENER – ITÔ DECOMPOSITION FOR THE IMAGE ... 749 J (φ) =   b0(φ) a∗0(φ) 0 0 0 . . . a0(φ) b1(φ) a∗1(φ) 0 0 . . . 0 a1(φ) b2(φ) a∗2(φ) 0 . . . ... ... ... ... ... . . .   with the entries an(φ) : Fn(H) → Fn+1(H), bn(φ) = (bn(φ))∗ : Fn(H) → Fn(H), a∗n(φ) = (an(φ))∗ : Fn+1(H) → Fn(H), φ ∈ H+, n ∈ Z+ = 0, 1, . . . . Each matrix J (φ) gives rise to a Hermitian operator J(φ) in the space F(H): Given a vector Φ = (Φn)∞n=0 ∈ Dom(J(φ)) = Ffin(H+), we define J(φ)Φ = ((J(φ)Φ)0, . . . , (J(φ)Φ)n, . . .) ∈ F(H), (J(φ)Φ)n = an−1(φ)Φn−1 + bn(φ)Φn + a∗n(φ)Φn+1, n ∈ Z+ (we suppose a−1(φ) = 0 and Φ−1 = 0). Consider the following assumptions. 1. The operators an(φ) and bn(φ), φ ∈ H+, n ∈ Z+, are bounded and real, i.e., they take real vectors to real ones. 2 (smoothness). The inclusions an(φ)(Fn(H+)) ⊂ Fn+1(H+), bn(φ)(Fn(H+)) ⊂ Fn(H+), a∗n(φ)(Fn+1(H+)) ⊂ Fn(H+), φ ∈ H+, n ∈ Z+, hold true. Basically, this axiom explains the relation between our family of operators and the fixed rigging. The tag “smoothness” appears because this axiom expresses the smooth- ness of coefficients when differential operators are considered for an(φ) and bn(φ). 3. The operators J(φ), φ ∈ H+, are essentially selfadjoint and their closures J̃(φ) are strongly commuting. 4. The functions H+ φ → an(φ)Φn ∈ Fn+1(H+), H+ φ → bn(φ)Φn ∈ Fn(H+), H+ φ → a∗n(φ)Φn+1 ∈ Fn(H+), n ∈ Z+, are linear and continuous for any Φn ∈ Fn(H+), Φn+1 ∈ Fn+1(H+). 5 (regularity). This last axiom expresses a rather complicated technical requirement. At the same time, it deals with a set of operators that are extremely important for our further constructions. ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6 750 Yu. M. BEREZANSKY, A. D. PULEMYOTOV The linear operators Vn : Fn(H+) → ⊕n j=0 Fj(H+) defined by the equalities V0 = IdC, Vn(φ1 ⊗̂ . . . ⊗̂ φn) = J(φ1) . . . J(φn)Ω, φ1, . . . , φn ∈ H+, n ∈ N, must be continuous. The operators Fn(H+) Fn → Vn,nFn = PrFn(H+) VnFn ∈ Fn(H+), n ∈ Z+, must be invertible. Again, we point out that Vn and Vn,n play a crucial role in the further considerations. The family J = (J̃(φ))φ∈H+ is called a (commutative) Jacobi field if Assumptions 1 – 5 are satisfied. (Recall that J̃(φ) stands for the closure of the operator J(φ).) Once again we should emphasize that the operators J̃(φ) act in the Fock space F(H). 2.2. Spectral theory of a Jacobi field. One can apply the projection spectral theo- rem (see [18, 29]) to the field J = (J̃(φ))φ∈H+ . We only adduce the result of such an application here. Proofs can be found in [2]. Given n ∈ Z+, let Pn stand for the set of all continuous polynomials H− ξ → n∑ j=0 〈ξ⊗j , aj〉H ∈ C, aj ∈ Fj(H+) (we suppose ξ⊗0 = 1.) The set P = ⋃∞ n=0 Pn is a dense subset of L2(H−, dρ). The closure of Pn in L2(H−, dρ) will be denoted by P̃n. Theorem 2.1. There exist a vector-valued function H− ξ → P (ξ) ∈ (Ffin(H+))′ and a Borel probability measure ρ on the space H− (the spectral measure) such that the following statements hold: 1. For every ξ ∈ H−, the vector P (ξ) ∈ (Ffin(H+))′ is a generalized joint eigenvec- tor of J with eigenvalue ξ, i.e., 〈P (ξ), J̃(φ)Φ〉H = 〈ξ, φ〉H〈P (ξ),Φ〉H , φ ∈ H+, Φ ∈ Ffin(H+). 2. The Fourier transform Ffin(H+) Φ → IΦ = 〈Φ, P (·)〉H ∈ L2(H−, dρ) can be extended to a unitary operator acting from F(H) to L2(H−, dρ). We preserve the notation I for this operator. 3. The Fourier transform I satisfies the equality IΦn = PrP̃n�P̃n−1 〈V −1 n,nΦn, ·⊗n〉H , Φn ∈ Fn(H+), n ∈ Z+ (we suppose P−1 = {0}). Corollary 2.1. The equality L2(H−, dρ) = ∞⊕ n=0 ( P̃n � P̃n−1 ) = ∞⊕ n=0 I(Fn(H+)) holds true. ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6 THE SPECTRAL THEORY AND THE WIENER – ITÔ DECOMPOSITION FOR THE IMAGE ... 751 Remark 2.1. If J is the classical free field, then Vn,n = √ n! IdFn(H+), n ∈ Z+, and ρ is the Gaussian measure. In this case, the Fourier transform I coincides with the Wiener – Itô – Segal transform and Corollary 2.1 constitutes the Wiener – Itô decomposi- tion for the Gaussian measure. Analogous results hold for the Poisson field. Remark 2.2. The equality IVnFn = 〈Fn, ·⊗n〉H , Fn ∈ Fn(H+), n ∈ Z+, holds true. See [22] for the proof. 3. Image of the spectral measure. This section aims to study the image of the mea- sure ρ under a bounded operator. Our main example for ρ will be the Poisson measure, as detailed in Section 4. We will now demonstrate how the image of ρ should be constructed for our purposes. Consider a real separable Hilbert space T. Let T− ⊃ T ⊃ T+ (3.1) be a rigging of T with real separable Hilbert spaces T+ and T− = (T+)′. As in the case of the rigging (2.1), the pairing in (3.1) can be extended to a pairing between Fn(T+) and Fn(T−). The latter can be extended to a pairing between Ffin(T+) and (Ffin(T+))′. We use the notation 〈·, ·〉T for all of these pairings. Consider a bounded operator K : T+ → H+ such that Ker(K) = {0}. We preserve the notation K for the extension of this operator to the complexified space (T+)C. The adjoint of K with respect to (2.1) and (3.1) is a bounded operator K+ : H− → T− defined by the equality 〈K+ξ, f〉T = 〈ξ,Kf〉H , ξ ∈ H−, f ∈ T+. One can prove that Ran(K+) is dense in T−. We denote by ρK the image of the measure ρ under the mapping K+. By definition, ρK is a probability measure on the σ-algebra C = { ∆ ⊂ T−\|(K+)−1(∆) is a Borel subset of H− } ((K+)−1(∆) denoting the preimage of the set ∆). Remark 3.1. Since the mapping K+ is Borel-measurable, the σ-algebra C contains the Borel σ-algebra of the space T−. If K+ takes Borel subsets of H− to the Borel subsets of T−, then C coincides with the Borel σ-algebra of T−. Remark 3.2. The characteristic functional ρ̂K(f) = ∫ T− ei〈ω,f〉T dρK(ω), f ∈ T+, of the measure ρK satisfies the equality ρ̂K(f) = ρ̂(Kf), f ∈ T+, with ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6 752 Yu. M. BEREZANSKY, A. D. PULEMYOTOV ρ̂(φ) = ∫ H− ei〈ξ,φ〉Hdρ(ξ), φ ∈ H+, being the characteristic functional of the measure ρ. Remark 3.3. The assumption Ker(K) = {0} is not essential. Indeed, the measure ρK proves to be lumped on the set of functionals which equal zero on Ker(K). This set can be naturally identified with (Ker(K)⊥)′. Thus we can always replace T+ with Ker(K)⊥ ⊂ T+. 3.1. The space Fext(T+, K) and its rigging. The goal is to produce a Fock-type space Fext(T+,K) and a family of operators in it associated to the measure ρK . We must mention the paper [30] that offers a very broad generalization of the classical Fock space and discusses families of operators in it. That theory seems to be closely related to what we suggest in this section. Before constructing Fext(T+,K), we have to introduce an auxiliary notation. Given n ∈ Z+, let Vn stand for the subspace of F(H) generated by the vectors VjK ⊗jFj , Fj ∈ Fj(T+), j = 0, . . . , n (we suppose K⊗0 = IdC). Introduce the mapping A : Ffin(T+) → F(H) via the formula Fn(T+) Fn → AFn = 1√ n! PrVn�Vn−1 VnK ⊗nFn ∈ F(H), n ∈ Z+ (we suppose V−1 = {0}). It is easy to see that Ker(A) = {0}. We use this mapping to identify the scalar product in our desired space Fext(T+,K). Having the required notation at hand, it is now possible to proceed with the definition. Let Fext n (T+,K) denote the completion of Fn(T+) with respect to the scalar product (Fn, Gn)Fext n (T+,K) = (AFn, AGn)F(H), Fn, Gn ∈ Fn(T+), n ∈ Z+. Put Fext(T+,K) = ∞⊕ n=0 Fext n (T+,K). This space has an evident Fock-type structure of an infinite orthogonal sum. Once can say that it was constructed on the basis of the Fock space F(T+). Obviously, the mapping A can be extended to an isometric operator acting from Fext(T+,K) to F(H). We preserve the notation A for this operator. Remark 3.4. If Ran(K) is dense in H+, then Vn = ⊕n j=0 Fj(H) and AFn = 1√ n! Vn,nK ⊗nFn, Fn ∈ Fn(T+), n ∈ Z+. If, additionally, J is the classical free field or the Poisson field, then AFn = K⊗nFn, Fn ∈ Fn(T+), n ∈ Z+, and the scalar product (·, ·)Fext(T+,K) satisfies the equality (Fn, Gn)Fext n (T+,K) = (K⊗nFn,K ⊗nGn)Fn(H), Fn, Gn ∈ Fn(T+), n ∈ Z+. This case has undergone a detailed study in [22]. ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6 THE SPECTRAL THEORY AND THE WIENER – ITÔ DECOMPOSITION FOR THE IMAGE ... 753 Remark 3.5. One can easily verify that A   n⊕ j=0 Fext j (T+,K)   = Vn, n ∈ Z+. Now we have to construct a rigging of the space Fext(T+,K). Recall that we are aiming for the spectral theory of the family JK . Clearly, the rigging of the space plays a crucial role in this pursuit. Our next definition may seem a little clumsy at the moment, but it will become natural when we present the explicit form of JK . Consider a linear topological space Fext + (T+,K) = A−1(Ffin(H+) ∩ Ran(A)). The sequence (Fn)∞n=0 converges to F in Fext + (T+,K) if and only if the sequence (AFn)∞n=0 converges to AF in Ffin(H+). The space Fext + (T+,K) is a dense subset of Fext(T+,K). Indeed, Ffin(H+) ∩ Vn is dense in Vn for all n ∈ Z+. Using Remark 3.5, we can conclude that Ffin(H+)∩Ran(A) is dense in Ran(A). Hence Fext + (T+,K) is dense in Fext(T+,K). Construct a rigging (Fext + (T+,K))′ ⊃ Fext(T+,K) ⊃ Fext + (T+,K). Denote the corresponding pairing by 〈·, ·〉A. 3.2. The family JK and its spectral theory. We will now define the family JK and prove the spectral theorem (an analogue of Theorem 2.1) for it. This would illuminate the connection between JK and ρK , thereby leading us to our goal. In particular, this would yield an analogue of the Wiener – Itô decomposition for the measure ρK . Generally, JK is not a Jacobi field, which is an important thing to understand. We begin with a simple technical lemma needed for the further definitions. Lemma 3.1. The set Ffin(H+)∩Ran(A) is invariant with respect to the operators J(Kf), f ∈ T+. Proof. According to the properties of a Jacobi field, the space Ffin(H+) is invariant with respect to J(Kf). Remark 3.5 implies that Ran(A) is generated by the vectors VnK ⊗nFn, Fn ∈ Fn(T+), n ∈ Z+. Evidently, J(Kf)VnK ⊗nFn = Vn+1K ⊗(n+1)(f ⊗̂ Fn) ∈ Ran(A), f ∈ T+. Since the operators J(Kf) � Fn(H) are bounded for any n ∈ Z+, we can conclude that Ffin(H+) ∩ Ran(A) is also invariant with respect to J(Kf). The lemma is proved. In the space Fext(T+,K), consider the operators JK(f) = A−1J(Kf)A, Dom(JK(f)) = Fext + (T+,K), f ∈ T+. Evidently, these operators are essentially selfadjoint and their closures J̃K(f) are strong commuting. ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6 754 Yu. M. BEREZANSKY, A. D. PULEMYOTOV Define JK = (J̃K(f))f∈T+ . Once again, JK is not a Jacobi field (at least, in general). Noteworthily, Lemma 3.1 implies that the space Fext + (T+,K) is invariant with respect to the operators J̃K(f), f ∈ T+. As before, we need to introduce the polynomials on our Hilbert space. Given n ∈ Z+, let Qn stand for the set of all continuous polynomials T− ω → n∑ j=0 〈ω⊗j , cj〉T ∈ C, cj ∈ Fj(T+) (we suppose ω⊗0 = 1). The closure of Qn in L2(T−, dρK) will be denoted by Q̃n. The spectral theorem for JK proceeds as follows. Theorem 3.1. Assume Q = ⋃∞ n=0 Qn to be a dense subset of L2(T−, dρK). There exists a vector-valued function T− ω → Q(ω) ∈ (Fext + (T+,K))′ such that the follow- ing statements hold: 1. For ρK-almost all ω ∈ T−, the vector Q(ω) ∈ (Fext + (T+,K))′ is a generalized joint eigenvector of the family JK with the eigenvalue ω, i.e., 〈Q(ω), J̃K(f)F 〉A = 〈ω, f〉T 〈Q(ω), F 〉A, F ∈ Fext + (T+,K). 2. The Fourier transform Fext + (T+,K) F → IKF = 〈F,Q(·)〉A ∈ L2(T−, dρK) can be extended to a unitary operator acting from Fext(T+,K) to L2(T−, dρK). We preserve the notation IK for this operator. 3. The Fourier transform IK satisfies the equality IKFn = 1√ n! PrQ̃n�Q̃n−1 〈Fn, ·⊗n〉T , Fn ∈ Fn(T+), n ∈ Z+ (3.2) (we suppose Q−1 = {0}). Introduce the isometric operator L2(T−, dρK) G → UG = G(K+·) ∈ L2(H−, dρ). Before proving Theorem 3.1, we have to state an auxiliary lemma. Lemma 3.2. The equality UQ̃n = IVn holds true. Proof. The space UQ̃n is generated by the vectors U〈Fj , ·⊗j〉T , Fj ∈ Fj(T+), j = 0, . . . , n. In virtue of Remark 2.2, U〈Fj , ·⊗j〉T = 〈K⊗jFj , ·⊗j〉H = IVjK ⊗jFj . Hence UQ̃n = IVn. The lemma is proved. Proof of Theorem 3.1. Let us construct the function Q(ω). Fix a dense subset (Fn)∞n=1 of the space Fext + (T+,K). According to the definition of I, IAFn = 〈AFn, P (·)〉H . ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6 THE SPECTRAL THEORY AND THE WIENER – ITÔ DECOMPOSITION FOR THE IMAGE ... 755 Lemma 3.2 and Remark 3.5 imply the equality Ran(IA) = Ran(U). Therefore we can find a set Ξ ⊂ H− and a sequence of functions (qn)∞n=1 ⊂ L2(T−, dρK) such that ρ(Ξ) = 1 and 〈AFn, P (ξ)〉H = qn(K+ξ), ξ ∈ Ξ, n ∈ N. (3.3) There exists a unique operator A+ : (Ffin(H+))′ → (Fext + (T+,K))′ such that 〈AFn, P (ξ)〉H = 〈Fn, A+P (ξ)〉A, ξ ∈ H−, n ∈ N. Formula (3.3) implies the equality A+P (ξ1) = A+P (ξ2) for the vectors ξ1, ξ2 ∈ Ξ such that K+ξ1 = K+ξ2. Define Q(K+ξ) = A+P (ξ) for ξ ∈ Ξ. We have constructed the function Q(ω) on the set K+Ξ ⊂ T−. Evidently, ρK(K+Ξ) = 1. We extend Q(ω) to the whole of the space T− arbitrarily. One can easily verify that IKF = 〈F,Q(·)〉A = U−1IAF, F ∈ Fext + (T+,K). The operator U−1IA is a unitary acting from Fext(T+,K) to L2(T−, dρK). Hence IK can be extended to a unitary operator acting from Fext(T+,K) to L2(T−, dρK). Let us show that Q(ω) is a generalized joint eigenvector of JK with the eigenvalue ω. Given F ∈ Fext + (T+,K), we obtain IK J̃K(f)F = U−1IAJ̃K(f)F = U−1IAA−1J̃(Kf)AF = = U−1IJ̃(Kf)AF = U−1〈J̃(Kf)AF,P (·)〉H = = U−1(〈Kf, ·〉H〈AF,P (·)〉H) = (U−1〈Kf, ·〉H)(U−1〈AF,P (·)〉H) = = 〈·, f〉T (U−1IAF ) = 〈·, f〉T (IKF ) (overbars denoting the complex conjugacy). The latter implies 〈Q(ω), J̃K(f)F 〉A = 〈ω, f〉T 〈Q(ω), F 〉A for ρK-almost all ω ∈ T−. To complete the proof, we have to show that IK satisfies equality (3.2). In accordance with Lemma 3.2 and Remark 2.2, IKFn = U−1IAFn = 1√ n! U−1I PrVn�Vn−1 VnK ⊗nFn = = 1√ n! U−1 PrIVn�IVn−1 IVnK ⊗nFn = = 1√ n! U−1 PrUQ̃n�UQ̃n−1 〈K⊗nFn, ·⊗n〉H = = 1√ n! U−1 PrUQ̃n�UQ̃n−1 U〈Fn, ·⊗n〉T = = 1√ n! PrQ̃n�Q̃n−1 〈Fn, ·⊗n〉T , Fn ∈ Fn(T+), n ∈ Z+. The theorem is proved. ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6 756 Yu. M. BEREZANSKY, A. D. PULEMYOTOV Corollary 3.1. If Q = ⋃∞ n=0 Qn is a dense subset of L2(T−, dρK), then the equal- ity L2(T−, dρK) = ∞⊕ n=0 ( Q̃n � Q̃n−1 ) = ∞⊕ n=0 IK(Fext n (T+,K)) hold true. Corollary 3.1 constitutes an analogue of the Wiener – Itô decomposition for the mea- sure ρK . 4. The operator of multiplication and the Poisson field. As mentioned in Section 1, it remains quite a challenge to carry out the calculations for a particular operator K and a particular field J. In this section, we illustrate the constructions of Section 3. We obtain an explicit formula for the scalar product (·, ·)Fext(T+,K) assuming K to be the operator of multiplication by a function of a new independent variable and J to be the Poisson field. This situation was considered in many works, and one of our objectives here is to give an operator theory approach to the problem. We hope this would be helpful in illuminating the complicated phenomena arising in the case under discussion. As we point out below, this case does not fall under the construction of [22]. Now Fext(T+,K) will not be a classical Fock space but rather an extended Fock space of a certain kind. Firstly, the abstract objects that appear in the previous section must be specified. Of course, T and H should now be function spaces. Consider a real separable Hilbert space S = L2(Rd1 , dσ). Let T equal L2(Rd2 , dτ). We assume the Borel measures σ and τ to be finite on compact sets. We also assume τ to be absolutely continuous with respect to the Lebesgue measure. Let the space H equal S ⊗ T. Clearly, H can be identified with L2(Rd1+d2 , d(σ ⊗ τ)). We choose the spaces T+ and H+ arbitrarily. Recall that the embedding H+ ↪→ H is assumed to be a Hilbert – Schmidt operator. Typically, the role of T+ and H+ is played by weighted Sobolev spaces. Define K via the formula T+ f(t) → (Kf)(s, t) = κ(s)f(t), s ∈ R d1 , t ∈ R d2 . (4.1) The function κ ∈ S has to be chosen so that K would be a bounded operator acting from T+ to H+. We emphasize that Ran(K) is not dense in H+ now, which prevents the example under discussion from being described by [22]. Suppose J = (J(φ))φ∈H+ to be the Poisson field in F(H). The operators of the Poisson field are defined via the formula J(φ) = J+(φ) + J0(φ) + J−(φ), φ ∈ H+. The operators J+(φ) and J−(φ) are the classical creation and annihilation operators in F(H), i.e., J+(φ)Fn = √ n + 1φ ⊗̂ Fn, φ ∈ H+, Fn ∈ Fn(H), n ∈ Z+, and J−(φ) = (J+(φ))∗. In order to define J0(φ), consider the operator b(φ) of multipli- cation by the function φ ∈ H+ in the space HC. We assume b(φ) to be bounded for any φ ∈ H+. For an arbitrary Fn ∈ Fn(H), define J0(φ)F0 = 0, ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6 THE SPECTRAL THEORY AND THE WIENER – ITÔ DECOMPOSITION FOR THE IMAGE ... 757 J0(φ)Fn = (b(φ) ⊗ IdH ⊗ . . .⊗ IdH)Fn + + (IdH ⊗b(φ) ⊗ IdH ⊗ . . .⊗ IdH)Fn + . . . . . . + (IdH ⊗ . . .⊗ IdH ⊗b(φ))Fn, φ ∈ H+, n ∈ N. In other words, J0(φ) equals the second (differential) quantization of b(φ). We assume J to satisfy the definition of a Jacobi field. The spectral measure ρ of the field J equals the centered Poisson measure with the intensity σ ⊗ τ. Its characteristic functional is given by the formula ρ̂(φ) = exp   ∫ Rd2 ∫ Rd1 ( eiφ(s,t) − 1 − iφ(s, t) ) dσ(s)dτ(t)  , φ ∈ H+. The operator K+ takes ρ to a probability measure ρK on T−. According to Remark 3.2, the characteristic functional of ρK is now given by the formula ρ̂K(f) = exp   ∫ Rd2 ∫ Rd1 ( eiκ(s)f(t) − 1 − iκ(s)f(t) ) dσ(s)dτ(t)  , f ∈ T+. Denote by σκ the image of σ under κ. The above formula implies that ρK is the Lévy noise measure on T− with the Lévy measure σκ and the intensity measure τ. 4.1. Preliminary constructions. Before identifying the scalar product (·, ·)Fext(T+,K) explicitly, we have to carry out some preliminary constructions. Basi- cally, we need to introduce all the ingredients in the formula for (·, ·)Fext(T+,K). Given n ∈ N, define κn(s) = (κ(s))n, s ∈ R d1 . Lemma 4.1. The function κn belongs to the space S for any n ∈ N. Proof. Consider a compact set ∆ ⊂ R d2 such that τ(∆) �= 0. Fix a smooth compactly supported function f0 ∈ T+ such that f0(t) = 1 for t ∈ ∆. In the space H, consider the bounded operator b(Kf0) of multiplication by Kf0 ∈ H+. The function (bn(Kf0)Kf0)(s, t) = κn+1(s)(f0(t))n+1, s ∈ R d1 , t ∈ R d2 , belongs to H for any n ∈ N. Hence the function kt0(s) = κn+1(s)(f0(t0))n+1 belongs to S for τ -almost all t0 ∈ ∆. Since kt0(s) = κn+1(s) for t0 ∈ ∆, the latter implies κn+1 ∈ S. The lemma is proved. Applying the Schmidt orthogonalization procedure to the sequence (κn)∞n=1, we ob- tain an orthogonal sequence (κn)∞n=0 in the space S. Each κn is a polynomial of degree n with respect to κ. We normalize κn so that the leading coefficient of this polynomial would equal 1. A vector F ∈ T⊗n C , n ∈ N, can be treated as a complex-valued function F (t1, . . . , tn) depending on the variables t1, . . . , tn ∈ R d2 . Analogously, a vector Φ ∈ H⊗n C , n ∈ N, can be treated as a complex-valued function Φ(s1, . . . , sn, t1, . . . , tn) depending on the variables s1, . . . , sn ∈ R d1 and t1, . . . , tn ∈ R d2 . Vectors from Fn(T ) and Fn(H) appear as symmetric functions. We assume the set of all smooth compactly supported functions on R d2n to be a dense subset of (T+)⊗n C . ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6 758 Yu. M. BEREZANSKY, A. D. PULEMYOTOV Consider an ordered partition ω = (ω1, . . . , ωk) of the set {1, . . . , n} into k nonempty sets ω1, . . . , ωk. Let Ωk n stand for the set of all such partitions and let \|ωk\| stand for the cardinality of ωk. Introduce the mapping R d2k (t1, . . . , tk) → πω(t1, . . . , tk) = (ti1 , . . . , tin) ∈ R d2n with ij = l for j ∈ ωl. 4.2. An explicit formula for the scalar product (·, ·)Fext(T+,K). We are now ready to identify the scalar product (·, ·)Fext(T+,K) explicitly. This will show the similarity be- tween Fext(T+,K) and the extended Fock space, on which we will elaborate after prov- ing the theorem. It is worthy to remark the relevance to the construction in [30]. Given a smooth compactly supported symmetric function F ∈ Fn(T+), n ∈ N, denote DF = (0, D1 F , . . . , Dn F , 0, 0, . . .) ∈ F(H) with Dk F (s1, . . . , sk, t1, . . . , tk) = = ∑ ω∈Ωk n 1√ k! ( κ\|ω1\|(s1) . . . κ\|ωk\|(sk) ) F (πω(t1, . . . , tk)), k = 1, . . . , n. Theorem 4.1. In the specific framework of this section, when K is the multipli- cation operator and J is the Poisson field, the scalar product (·, ·)Fext(T+,K) can be computed explicitly. It satisfies the equality (F,G)Fext(T+,K) = 1 n! n∑ k=1 ∫ Rd2k ∫ Rd1k Dk F (s, t)Dk G(s, t) dσ⊗k(s)dτ⊗k(t) (4.2) for any smooth compactly supported symmetric functions F,G ∈ Fn(T+), n ∈ N (the overbar denoting the complex conjugacy). This describes the space Fext(T+,K) and leads to the Wiener – Itô decomposition for ρK . Before proving Theorem 4.1, we have to introduce some notations and make some remarks. We also have to state three auxiliary lemmas. According to the definition of (·, ·)Fext(T+,K), it suffices to show that AF = 1√ n! DF . Calculating AF involves identifying the subspaces Vl, l ∈ Z+, and the vector VnK ⊗nF. We represent Vl as the subspace of F(H) generated by an explicitly described set of functions. More precisely, fix an orthonormal basis (ci)∞i=1 of the space T such that each function ci is smooth and compactly supported. Define the symmetrization operator ˆ on T⊗k C and H⊗k C , k ∈ N, via the formulas F̂ (t1, . . . , tk) = 1 k! ∑ ι∈Sk F ( tι(1), . . . , tι(k) ) , F ∈ T⊗k C , Φ̂(s1, . . . , sk, t1, . . . , tk) = 1 k! ∑ ι∈Sk Φ ( sι(1), . . . , sι(k), tι(1), . . . , tι(k) ) , Φ ∈ H⊗k C (Sk denoting the group of all permutations of the set {1, . . . , k}). Given l ∈ N, let Wl stand for the subspace of F(H) generated by the vacuum Ω and the functions (κi1(s1)cj1(t1) . . . κik (sk)cjk (tk))̂ , k, i1, . . . , ik, j1, . . . , jk ∈ N, i1 + . . . + ik ≤ l. ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6 THE SPECTRAL THEORY AND THE WIENER – ITÔ DECOMPOSITION FOR THE IMAGE ... 759 We suppose W0 = C. Lemma 4.2 establishes the necessary properties of the subspaces Wl. Lemma 4.4 shows that Vl = Wl for any l ∈ Z+. It should be noted that the proof of Lemma 4.4 is based on the arguments from [24]. We represent the vector VnK ⊗nF as the sum of an explicitly described vector BF and a vector NF ∈ Wn−1. Lemma 4.3 establishes the corresponding formula. When put together, the assertions of Lemma 4.3 and Lemma 4.4 enable us to show that AF = 1√ n! DF . As mentioned above, this completes the proof of Theorem 4.1. Lemma 4.2. The inclusions J+(Kci)Wl ⊂ Wl+1, J0(Kci)Wl ⊂ Wl+1, J−(Kci)Wl ⊂ Wl−1, i ∈ N, hold true. The first and the second inclusion hold for l ∈ Z+. The third one holds for l ∈ N. Proof. Let us prove the first inclusion. The second and the third one can be proved analogously. Evidently, J+(Kci)Ω ∈ Wl+1 for any l ∈ Z+. By calculating the function J+(Kci)(κi1(s1)cj1(t1) . . . κik (sk)cjk (tk))̂ , k, i1, . . . , ik, j1, . . . , jk ∈ N, explicitly, one can show that J+(Kci)(κi1(s1)cj1(t1) . . . κik (sk)cjk (tk))̂ ∈ Wl+1 as soon as i1 + . . . + ik ≤ l. Furthermore, the restriction J+(Kci) � Wl is a bounded operator because Wl ⊂ ⊕l j=0 Fj(H). Thus J+(Kci)Wl ⊂ Wl+1 for any l ∈ Z+. The lemma is proved. Denote BF = (0, B1 F , . . . , Bn F , 0, 0, . . .) ∈ F(H) with Bk F (s1, . . . , sk, t1, . . . , tk) = = ∑ ω∈Ωk n 1√ k! ( κ\|ω1\|(s1) . . . κ\|ωk\|(sk) ) F (πω(t1, . . . , tk)), k = 1, . . . , n. Observe that although BF resembles DF , these vectors are not identical. Lemma 4.3. The equality VnK ⊗nF = BF + NF , holds with NF ∈ Wn−1. Proof. The statement is evident for n = 1. Assume it to hold for n = m. Fix the basis vectors cj1 , . . . , cjm+1 . Denote Fm(t1, . . . , tm) = (cj1(t1) . . . cjm (tm))̂ . It suffices to carry out the proof for the function Fm+1(t1, . . . , tm+1) = (cjm+1(tm+1)Fm(t1, . . . , tm))̂ . According to the induction hypothesis, ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6 760 Yu. M. BEREZANSKY, A. D. PULEMYOTOV Vm+1K ⊗m+1Fm+1 = J(Kcjm+1)VmK⊗mFm = = J+(Kcjm+1)BFm + J0(Kcjm+1)BFm + J−(Kcjm+1)BFm + J(Kcjm+1)NFm . A straightforward calculation shows that J+(Kcjm+1)BFm + J0(Kcjm+1)BFm = BFm+1 . Define NFm+1 = J−(Kcjm+1)BFm + J(Kcjm+1)NFm . One can easily verify that BFm ∈ Wm. Furthermore, NFm ∈ Wm−1 by the induction hypothesis. Hence, according to Lemma 4.2, the vector NFm+1 ∈ Wm. The lemma is proved. Lemma 4.4. The equality Vl = Wl holds for any l ∈ Z+. Proof. The statement is evident for l = 0. Assume it to hold for l = m. The inclusion Vm+1 ⊂ Wm+1 is a direct consequence of Lemma 4.3. In order to prove the converse inclusion, consider the function G(s1, . . . , sk, t1, . . . , tk) = κi1(s1) . . . κik(sk)g(t1, . . . , tk), i1 + . . . + ik = m + 1, k, i1, . . . , ik ∈ N, with g being smooth and compactly supported. It suffices to prove that Ĝ ∈ Vm+1. In order to do this, we will now construct a sequence (Gj)∞j=1 ⊂ Vm+1 which converges to Ĝ in the space F(H). Given a number j ∈ N and a partition ω ∈ Ωi m+1 with i = 1, . . . ,m + 1, introduce a smooth function hω,j on R d2(m+1) satisfying the following requirements: 1) the estimate 0 ≤ hω,j(t) ≤ 1 holds for all t ∈ R d2(m+1); 2) the equality hω,j(t) = 1 holds for t ∈ Ran(πω); 3) the equality hω,j(t) = 0 holds for all t such that the distance between t and Ran(πω) is greater than 1 j . The existence of hω,j is easy to prove. Fix ω′ ∈ Ωk m+1 such that \|ω′ 1\| = i1, . . . , \|ω′ k\| = ik. Introduce the functions Γj(t) = g ( π−1 ω′ ( PrRan(πω′ ) t )) hω′,j(t) k−1∏ i=1 ∏ ω∈Ωi m+1 (1 − hω,j(t)), t ∈ R d2(m+1), j ∈ N. According to Lemma 4.3, the equality Vm+1K ⊗(m+1)Γ̂j = BΓ̂j + NΓ̂j , j ∈ N, holds with NΓ̂j ∈ Wm = Vm. Define Gj = 1√ k! BΓ̂j . To complete the proof, we have to show that (Gj)∞j=1 converges to Ĝ in the space F(H). ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6 THE SPECTRAL THEORY AND THE WIENER – ITÔ DECOMPOSITION FOR THE IMAGE ... 761 Clearly, B1 Γ̂j = . . . = Bk−1 Γ̂j = 0. Given k0 ≥ k, consider a partition ω ∈ Ωk0 m+1. If ω = ( ω′ ι(1), . . . , ω ′ ι(k) ) for some ι ∈ Sk, then lim j→∞ ∥∥∥κ\|ω1\|(s1) . . . κ\|ωk\|(sk)Γj(πω(t1, . . . , tk))− − G ( sι−1(1), . . . , sι−1(k), tι−1(1), . . . , tι−1(k) )∥∥∥ Fk(H) = 0. Otherwise, lim j→∞ ∥∥∥κ\|ω1\|(s1) . . . κ\|ωk0 \|(sk0)Γj(πω(t1, . . . , tk0)) ∥∥∥ Fk0 (H) = 0. Hence lim j→∞ ‖Gj − Ĝ‖F(H) = lim j→∞ ∥∥∥∥ 1√ k! Bk Γ̂j − Ĝ ∥∥∥∥ Fk(H) = 0 and (Gj)∞j=1 converges to Ĝ in the space F(H). The lemma is proved. Proof of Theorem 4.1. According to the definition of (·, ·)Fext(T+,K), it suffices to prove that AF = 1√ n! DF or, equivalently, VnK ⊗nF −DF = PrVn−1 VnK ⊗nF. (4.3) Lemma 4.3 implies the equality VnK ⊗nF −DF = BF + NF −DF with NF ∈ Wn−1. One can easily see that ( Bk F −Dk F ) (s1, . . . , sk, t1, . . . , tk) = = ∑ ω∈Ωk n 1√ k! ( κ\|ω1\|(s1) . . . κ\|ωk\|(sk) − κ\|ω1\|(s1) . . . κ\|ωk\|(sk) ) × ×F (πω(t1, . . . , tk)) ∈ Wn−1 for any k = 1, . . . , n. Therefore (VnK ⊗nF −DF ) ∈ Wn−1 = Vn−1. To finish the proof of (4.3), we have to show that the difference VnK ⊗nF − (VnK ⊗nF −DF ) = DF is orthogonal to Vn−1. Since Vn−1 = Wn−1, it suffices to show that Dk F is orthogonal to the function ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6 762 Yu. M. BEREZANSKY, A. D. PULEMYOTOV G(s1, . . . , sk, t1, . . . , tk) = (κi1(s1)cj1(t1) . . . κik (sk)cjk (tk))̂ , i1 + . . . + ik ≤ n− 1, i1, . . . , ik, j1, . . . , jk ∈ N, for any k = 1, . . . , n− 1. The scalar product (Dk F , G)Fk(H) can be represented as a linear combination of the expressions ( κ\|ω1\|, κiι(1) ) S . . . ( κ\|ωk\|, κiι(k) ) S , ι ∈ Sk. Using the inequality \|ω1\| + . . . + \|ωk\| = n > n− 1 ≥ i1 + . . . + ik and the orthogonality of κi, one can easily prove that each of these expressions equals 0. Thus DF is orthogonal to Vn−1. The theorem is proved. Theorems 3.1 and 4.1 explain the Fock-type structure of L2(T−, dρK). Theorem 4.1 shows that the space Fext(T+,K) coincides with the extended Fock space investigated in [11] up to scalar weights at the orthogonal components. Using the formula AF = 1√ n! DF , one can construct an embedding of Fext(T+,K) into a weighted orthogonal sum of function spaces. Using the arguments from [24], see also [7, 8, 10 – 12], one can extend this imbedding to a unitary operator. We do not adduce the details of the corresponding construction here. Let us carry out a more accurate comparison of the results of the present paper with the results of [11]. As before, we suppose K to be defined by (4.1) and J to be the Poisson field. The extended Fock space is defined in [11] as the weighted orthogonal sum F = ∞⊕ n=0 Fn n!. In this formula, the space Fn coincides with the completion of Fn(T+) with respect to the scalar product given by the right-hand side of (4.2) (the space F0 = C.) Theorem 4.1 of the present paper implies the equality Fn = Fext n (T+,K). Corollary 5.3 of [11] yields a unitary operator U acting from F to L2(T−, dρK). Noteworthily, the unitarity of U is proved by confronting U with an operator I acting from an auxiliary Hilbert space H to L2(T−, dρK) as a sum of operators of stochastic integration. Theorem 4.1 and The- orem 5.1 of [11] establish the crucial properties of I. Theorem 3.1 of the present paper implies that U coincides with IK on Fext n (T+,K) up to the scalar factor 1√ n! . Acknowledgement. The authors express their gratitude to Prof. Leonard Gross and Dr. Eugene Lytvynov for taking the time to discuss and critique the paper. 1. Berezansky Yu. M. Direct and inverse spectral problems for Jacobi fields // St.Petersburg Math. J. – 1998. – 9. – P. 1053 – 1071. 2. Berezansky Yu. M. Commutative Jacobi fields in Fock space // Integr. Equat. Oper. Theory. – 1998. – 30. – P. 163 – 190. 3. Berezansky Yu. M. On the theory of commutative Jacobi fields // Meth. Funct. Anal. and Top. – 1998. – 4, # 1. – P. 1 – 31. 4. Berezansky Yu. M. Spectral theory of commutative Jacobi fields: Direct and inverse problems // Fields Inst. Commun. – 2000. – 25. – P. 211 – 224. ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6 THE SPECTRAL THEORY AND THE WIENER – ITÔ DECOMPOSITION FOR THE IMAGE ... 763 5. Lytvynov E. W. Multiple Wiener integrals and non-Gaussian white noises: A Jacobi field approach // Meth. Funct. Anal. and Top. – 1995. – 1, # 1. – P. 61 – 85. 6. Berezansky Yu. M. Poisson measure as the spectral measure of Jacobi field // Infin. Dim. Anal. Quant. Probab. Relat. Top. – 2000. – 3. – P. 121 – 139. 7. Kondratiev Yu. G., Lytvynov E. W. Operators of gamma white noise calculus // Ibid. – P. 303 – 335. 8. Berezansky Yu. M., Mierzejewski D. A. The chaotic decomposition for the gamma field // Funct. Anal. and Appl. – 2001. – 35. – P. 263 – 266. 9. Berezansky Yu. M., Lytvynov E. W., Mierzejewski D. A. The Jacobi field of a Lévy process // Ukr. Math. J. – 2003. – 55, # 5. – P. 706 – 710. 10. Berezansky Yu. M., Mierzejewski D. A. The construction of chaotic representation for the gamma field // Infin. Dim. Anal. Quant. Probab. Relat. Top. – 2003. – 6. – P. 33 – 56. 11. Lytvynov E. W. Orthogonal decompositions for Lévy processes with an application to the gamma, Pascal, and Meixner processes // Ibid. – P. 73 – 102. 12. Lytvynov E. W. Polynomials of Meixner’s type in infinite dimensions – Jacobi fields and orthogonality measures // J. Funct. Anal. – 2003. – 200. – P. 118 – 149. 13. Lytvynov E. W. The square of white noise as a Jacobi field // Infin. Dim. Anal. Quant. Probab. Relat. Top. – 2004. – 7. – P. 619 – 629. 14. Krein M. G. Infinite J-matrices and matrix-moment problem // Dokl. Akad. Nauk SSSR. – 1949. – 69, # 2. – P. 125 – 128 (in Russian). 15. Gekhtman M. I., Kalyuzhny A. A. Spectral theory of orthogonal polynomials in several variables // Ukr. Math. J. – 1991. – 43, # 10. – P. 1334 – 1337. 16. Gekhtman M. I., Kalyuzhny A. A. On the orthogonal polynomials in several variables // Integr. Equat. Oper. Theory. – 1994. – 19. – P. 404 – 418. 17. Dunkl C. F., Yuan Xu. Orthogonal polynomials of several variables // Encycl. Math. and Appl. – 2001. – 81. 18. Berezansky Yu. M., Kondratiev Yu. G. Spectral methods in infinite-dimensional analysis. – Dordrecht: Kluwer, 1995 (in Russian: Kiev: Naukova Dumka, 1988). 19. Hudson R. L., Parthasarathy K. R. Quantum Itô’s formula and stochastic evolutions // Communs Math. Phys. – 1984. – 93. – P. 301 – 323. 20. Surgailis D. On multiple Poisson stochastic integrals and associated Markov semigroups // Probab. Math. Statist. – 1984. – 3. – P. 217 – 239. 21. Janson S. Gaussian Hilbert spaces. – Cambridge: Cambridge Univ. Press, 1997. 22. Berezansky Yu. M., Lytvynov E. W., Pulemyotov A. D. Image of the spectral measure of a Jacobi field and the corresponding operators // Integr. Equat. Oper. Theory. – 2005. – 53. – P. 191 – 208. 23. Kondratiev Yu. G., Silva J. L., Streit L., Us G. F. Analysis on Poisson and Gamma spaces // Infin. Dim. Anal. Quant. Probab. Relat. Top. – 1998. – 1. – P. 91 – 117. 24. Berezansky Yu. M., Mierzejewski D. A. The structure of the extended symmetric Fock space // Meth. Funct. Anal. and Top. – 2000. – 6, # 4. – P. 1 – 13. 25. Lee Y.-J., Shih H.-H. The Segal – Bargmann transform for Lévy functionals // J. Funct. Anal. – 1999. – 168. – P. 46 – 83. 26. Nualart D., Schoutens W. Chaotic and predictable representations for Lévy processes // Stochast. Process. and Appl. – 2000. – 90. – P. 109 – 122. 27. Schoutens W. Stochastic processes and orthogonal polynomials // Lect. Notes Statist. – 2000. – 146. 28. Tsilevich N., Vershik A. Fock factorizations, and decompositions of the L2 spaces over general Lévy processes // Rus. Math. Surv. – 2003. – 58, # 3. – P. 427 – 472. 29. Pulemyotov A. D. Support of a joint resolution of identity and the projection spectral theorem // Infin. Dim. Anal. Quant. Probab. Relat. Top. – 2003. – 6. – P. 549 – 561. 30. Mierzejewski D. A. Generalized Jacobi fields // Meth. Funct. Anal. and Top. – 2003. – 9, # 1. – P. 80 – 100. Received 13.12.2006 ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6

Spectral theory and Wiener-Itô decomposition for the image of a Jacobi field

Institution

Ähnliche Einträge