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An equivalent representation of the Jacobi field of a Lévy process 5 0 E. Lytvynov 0 2 Department of Mathematics n University of Wales Swansea a J Singleton Park 5 Swansea SA2 8PP 2 U.K. ] E-mail: [email protected] R P Dedicated to Professor Yuri Berezansky . h on his 80th birhday t a m Abstract [ In [8], the Jacobi field of a Lévy process was derived. This field consists 1 v of commuting self-adjoint operators acting in an extended (interacting) Fock 0 space. However, these operators have a quite complicated structure. In this 5 note, using ideas from [1, 17], we obtain a unitary equivalent representation 4 1 of the Jacobi field of a Lévy process. In this representation, the operators act 0 in a usual symmetric Fock space and have a much simpler structure. 5 0 AMS Mathematics Subject Classification: 60G20, 60G51, 60H40, 47B36 / h t a m 1 Lévy process and its Jacobi field : v i ThenotionofaJacobifieldintheFockspace first appearedintheworks byBerezan- X sky and Koshmanenko [6, 7], devoted to the axiomatic quantum field theory, and r a then was further developed by Bru¨ning (see e.g. [10]). These works, however, did not contain any relations with probability measures. A detailed study of a general commutative Jacobi field in the Fock space and a corresponding spectral measure was carried out in a serious of works by Berezansky, see e.g. [3, 5] and the references therein. In [8] (see also [16, 18]), the Jacobi field of a Lévy process on a general manifold X was studied. Let us shortly recall these results. Let X be a complete, connected, oriented C∞ (non-compact) Riemannian manifold and let B(X) be the Borel σ-algebra on X. Let σ be a Radon measure on (X,B(X)) that is non-atomic and non-degenerate (i.e., σ(O) > 0 for any open set O ⊂ X). As a typical example of measure σ, one can take the volume measure on X. 1 We denote by D the space C∞(X) of all infinitely differentiable, real-valued 0 functions on X with compact support. It is known that D can be endowed with a topology of a nuclear space. Thus, we can consider the standard nuclear triple D ⊂ L2(X,σ) ⊂ D′, where D′ is the dual space of D with respect to the zero space L2(X,σ). (Here and below, all the linear spaces we deal with are real.) The dual pairing between ω ∈ D′ and ϕ ∈ D will be denoted by hω,ϕi. We denote the cylinder σ-algebra on D′ by C(D′). Let ν be a measure on (R,B(R)) whose support contains an infinite number of points and assume ν({0}) = 0. Let ν˜(ds) := s2ν(ds). We further assume that ν˜ is a finite measure on (R,B(R)), and moreover, there exists an ε > 0 such that exp ε|s| ν˜(ds) < ∞. (1) ZR (cid:0) (cid:1) We now define a centered Lévy process on X (without Gaussian part) as a generalized process on D′ whose law is the probability measure µ on (D′,C(D′)) given by its Fourier transform eihω,ϕiµ(dω) = exp (eisϕ(x) −1−isϕ(x))ν(ds)σ(dx) , ϕ ∈ D. (2) ZD′ (cid:20)ZR×X (cid:21) Thus, ν is the Lévy measure of the Lévy process µ. Without loss of generality, we can suppose that ν˜ is a probability measure on R. (Indeed, if this is not the case, define ν′ := c−1ν and σ′ := cσ, where c := ν˜(R).) It follows from (1) that the measure ν˜ has all moments finite, and furthermore, the set of all polynomials is dense in L2(R,ν˜). Therefore, by virtue of [2], there exists a unique (infinite) Jacobi matrix a b 0 0 0 ... 0 1 b a b 0 0 ... 1 1 2 J = 0 b2 a2 b3 0 ..., an ∈ R, bn > 0, 0 0 b a b ...  3 3 4   ... ... ... ... ... ...   whose spectral measure is ν˜. Next, we denote by P(D′) the set of continuous polynomials onD′, i.e., functions on D′ of the form F(ω) = n hω⊗i,f i, ω⊗0:=1, f ∈ D⊗î, i = 0,...,n, n ∈ Z . i=0 i i + P 2 Here, ⊗ˆ stands for symmetric tensor product. The greatest number i for which f(i) 6= 0 is called the power of a polynomial. We denote by P (D′) the set of n continuous polynomials of power ≤ n. By (1), (2), and [19, Sect. 11], P(D′) is a dense subset of L2(D′,µ). Let P∼(D′) n denote the closure of P (D′) in L2(D′,µ), let P (D′), n ∈ N, denote the orthogonal n n difference P∼(D′)⊖P∼ (D′), and let P (D′):=P∼(D′). Then, we evidently have: n n−1 0 0 ∞ L2(D′,µ) = P (D′). (3) n Mn=0 The set of all projections :h·⊗n,f i: of continuous monomials h·⊗n,f i, f ∈ D⊗ˆn, n n n onto P (D′) is dense in P (D′). For each n ∈ N, we define a Hilbert space F as n n n the closure of the set D⊗ˆn in the norm generated by the scalar product 1 (f ,g ) := :hω⊗n,f i::hω⊗n,g i:µ(dω), f ,g ∈ D⊗ˆn. (4) n n Fn n! Z n n n n D′ Denote ∞ F:= F n!, (5) n Mn=0 where F :=R. By (3)–(5), we get the unitary operator 0 U : F → L2(D′,µ) that is defined through Uf :=:h·⊗n,f i:, f ∈ D⊗ˆn, n ∈ Z , and then extended by n n n + linearity and continuity to the whole space F. An explicit formula for the scalar product (·,·) looks as follows. We denote by Fn Z∞ the set of all sequences α of the form +,0 α = (α ,α ,...,α ,0,0,...), α ∈ Z , n ∈ N. 1 2 n i + Let |α|:= ∞ α . For each α ∈ Z∞ , 1α + 2α + ··· = n, n ∈ N, and for any i=1 i +,0 1 2 function fP: Xn → R we define a function D f : X|α| → R by setting n α n (D f )(x ,...,x ):=f(x ,...,x ,x ,x ,x ,x ,...,x ,x , α n 1 |α| 1 α1 α1+1 α1+1 α1+2 α1+2 α1+α2 α1+α2 2times 2times 2times x ,x| {z,x } | ,.{.z.). } | {z } α1+α2+1 α1+α2+1 α1+α2+1 3times | {z } We have (cf. [16]): 3 Theorem 1 For any f(n),g(n) ∈ D⊗ˆn, we have: (f(n),g(n)) = K (D f )(x ,...,x ) Fn αZ α n 1 |α| α∈Z∞+,0:1αX1+2α2+···=n X|α| ×(D g )(x ,...,x )σ⊗|α|(dx ,...,dx ), α n 1 |α| 1 |α| where (1α +2α +···)! k−1b 2αk K = 1 2 i=1 i . (6) α α !α !··· (cid:18)Qk! (cid:19) 1 2 kY≥2 Next, we find the elements which belong to the space F after the completion of n D⊗ˆn. To this end, we define, for each α ∈ Z∞ , the Hilbert space +,0 L2(X|α|,σ⊗|α|):=L2(X,σ)⊗ˆα1 ⊗L2(X,σ)⊗ˆα2 ⊗··· . α Define a mapping U(n) : D⊗ˆn → L2(X|α|,σ⊗|α|)K α α α∈Z∞+,0:1Mα1+2α2+···=n by setting, for each f(n) ∈ D⊗ˆn, the L2(X|α|,σ⊗|α|)K -coordinate of U(n)f(n) to be α α D f(n). By virtue of Theorem 1, U(n) may be extended by continuity to anisometric α mapping of F into n L2(X|α|,σ⊗|α|)K . α α α∈Z∞+,0:1Mα1+2α2+···=n Furthermore, we have (cf. [9, 16]): Theorem 2 The mapping U(n) : F → L2(X|α|,σ⊗|α|)K n α α α∈Z∞+,0:1Mα1+2α2+···=n is a unitary opertator. By virtue of Theorem 2 and (5), we can identify F with the space n L2(X|α|,σ⊗|α|)K α α α∈Z∞+,0:1Mα1+2α2+···=n and the space F with L2(X|α|,σ⊗|α|)K (1α +2α +···)!. α α 1 2 α∈MZ∞ +,0 4 For a vector f ∈ F, we will denote its α-coordinate by f . α Note that, for for α = (n,0,0,...), we have L2(X|α|,σ⊗|α|) = L2(X,σ)⊗ˆn, K = 1, (1α +2α +···)! = n!. α α 1 2 Hence, the space F contains the symmetric Fock space ∞ F(L2(X,σ)) = L2(X,σ)⊗ˆnn! Mn=0 as a proper subspace. Therefore, we call F an extended Fock space. We also note that the space F satisfies the axioms of an interacting Fock space, see [11]. In the space L2(D′,µ), we consider, for each ϕ ∈ D, the operator M(ϕ) of multiplication by the function h·,ϕi. Let J(ϕ) := UM(ϕ)U−1. Denote by F (D) fin the set of all vectors of the form (f ,f ,...,f ,0,0,...), f ∈ D⊗î, i = 0,...,n, 0 1 n i n ∈ Z . Evidently, F (D) is a dense subset of F. We have the following theorem, + fin see [8]. Theorem 3 For any ϕ ∈ D, we have: F (D) ⊂ Dom(J(ϕ)), J(ϕ) ↾ F (D) = J+(ϕ)+J0(ϕ)+J−(ϕ). (7) fin fin Here, J+(ϕ) is the usual creation operator: J+(ϕ)f = ϕ⊗ˆf , f ∈ D⊗ˆn, n ∈ Z . (8) n n n + Next, for each f(n) ∈ D⊗ˆn, J0(ϕ)f(n) ∈ F and n (J0(ξ)f(n)) (x ,...,x ) α 1 |α| ∞ = α a S ξ(x )(D f(n))(x ,...,x ) k k−1 α α1+···+αk α 1 |α| Xk=1 (cid:0) (cid:1) σ⊗|α|-a.e., α ∈ Z∞ , 1α +2α +··· = n, (9) +,0 1 2 J−(ξ)f(n) = 0 if n = 0, J−(ξ)f(n) ∈ F if n ∈ N and n−1 (J−(ξ)f(n)) (x ,...,x ) α 1 |α| = nS ξ(x)(D f(n))(x,x ,...,x )σ(dx) α(cid:18)Z α+11 1 |α| (cid:19) X n + α b2 S ξ(x )(D f(n))(x ,...,x ) k k−1 k−1 α α1+···+αk α−1k−1+1k 1 |α| Xk≥2 (cid:0) (cid:1) σ⊗|α|-a.e., α ∈ Z∞ , 1α +2α +··· = n−1. (10) +,0 1 2 5 In formulas (9) and (10), we denoted by S the orthogonal projection of α L2(X|α|,σ⊗|α|) onto L2(X|α|,σ⊗|α|), α α±1 :=(α ,...,α ,α ±1,α ,...), α ∈ Z∞ , n ∈ N. n 1 n−1 n n+1 +,0 Finally, each operator J(ϕ), ϕ ∈ D, is essentially self-adjoint on F (D). fin By (7), theoperatorJ(ϕ) ↾ F (D)isasumofcreation, neutral, andannihilation fin operators, and hence J(ϕ) ↾ F (D) has a Jacobi operator’s structure. The family fin of operators (J(ϕ)) is called the Jacobi field corresponding to the Lévy process ϕ∈D µ. 2 An equivalent representation As shown in [4, 13, 15, 16], in some cases, the formulas describing the operators J0(ϕ) and J−(ϕ) can be significantly simplified. However, in the case of a general Lévy process this is not possible, see [8]. We will now present a unitarily equivalent description of the Jacobi field (J(ϕ)) , which will have a simpler form. ϕ∈D Let us consider the Hilbert space ℓ spanned by the orthonormal basis (e )∞ 2 n n=0 with e = (0,...,0, 1 ,0,0...). n (n+1)-stplace |{z} Consider the tensor product ℓ ⊗L2(X,σ), and let 2 ∞ F(ℓ ⊗L2(X,σ)) = (ℓ ⊗L2(X,σ))⊗ˆnn! 2 2 Mn=0 be the (usual) symmetric Fock space over ℓ ⊗L2(X,σ). 2 Denote by ℓ the dense subset of ℓ consisting of all finite vectors, i.e., 2,0 2 ℓ :={(f(n))∞ : ∃N ∈ Z such that f(n) = 0 for all n ≥ N}. 2,0 n=0 + The Jacobi matrix J determines a linear symmetric operator in ℓ with domain ℓ 2 2,0 by the following formula: Je = b e +a e +b e , n ∈ Z , e :=0. (11) n n+1 n+1 n n n n−1 + −1 DenotebyJ+,J0,J− thecorrespondingcreation, neutral,andannihilationoperators in ℓ , so that J = J+ +J0 +J−. 2,0 Denote by Φ the linear subspace of F(ℓ ⊗L2(X,σ)) that is the linear span of 2 the vacuum vector (1,0,0,...) and vectors of the form (ξ ⊗ ϕ)⊗n, where ξ ∈ ℓ , 2,0 ϕ ∈ D, and n ∈ N. The set Φ is evidently a dense subset of F(ℓ ⊗L2(X,σ)). 2 6 Now, for each ϕ,ψ ∈ D and ξ ∈ ℓ , we set 2,0 A+(ϕ)(ξ ⊗ψ)⊗n:=(e ⊗ϕ)⊗ˆ(ξ ⊗ψ)⊗n 0 +n((J+ξ)⊗(ϕψ))⊗ˆ(ξ ⊗ϕ)⊗(n−1), A0(ϕ)(ξ ⊗ψ)⊗n:=n((J0ξ)⊗(ϕψ))⊗ˆ(ξ ⊗ϕ)⊗(n−1), A−(ϕ)(ξ ⊗ψ)⊗n:=nhξ,e ihϕ,ψi(ξ ⊗ϕ)⊗(n−1) 0 +n((J−ξ)⊗(ϕψ))⊗ˆ(ξ ⊗ϕ)⊗(n−1), (12) and then extend these operators by linearity to the whole Φ. Thus, A+(ϕ) = a+(e ⊗ϕ)+a0(J+ ⊗ϕ), 0 A0(ϕ) = a0(J0 ⊗ϕ), A−(ϕ) = a−(e ⊗ϕ)+a0(J− ⊗ϕ), 0 where a+(·), a0(·), a−(·) are the usual creation, neutral, and annihilation operators in F(ℓ ⊗L2(X,σ)). (In fact, under, e.g., a0(J+⊗ϕ) we understand the differential 2 second quantization of the operator J+ ⊗ ϕ in ℓ ⊗ L2(X,σ), which, in turn, is 2 the tensor product of the operator J+ in ℓ defined on ℓ and the operator of 2 2,0 multiplication by ϕ in L2(X,σ) defined on D.) Note also that A(ϕ):=A+(ϕ)+A0(ϕ)+A−(ϕ) =a+(e ⊗ϕ)+a0((J+ +J0 +J−)⊗ϕ)+a−(e ⊗ϕ) 0 0 =a+(e ⊗ϕ)+a0(J ⊗ϕ)+a−(e ⊗ϕ). 0 0 In the following theorem, for a linear operator A in a Hilbert space H, we denote by A its closure (if it esxists). Theorem 4 There exists a unitary operator I : F → F(ℓ ⊗L2(X,σ)) 2 for which the following assertions hold. Let J+(ϕ), J0(ϕ), and J−(ϕ), ϕ ∈ D, be linear operators in F with domain F (D) as in Theorem 3. Then, for all ϕ ∈ D, fin IJ+(ϕ)I−1 = A+(ϕ), IJ0(ϕ)I−1 = A0(ϕ), IJ−(ϕ)I−1 = A−(ϕ), and IJ(ϕ)I−1 = A(ϕ). 7 Remark 1 Note, however, that the image of F under I does not coincide with the n subspace (ℓ ⊗L2(X,σ))⊗ˆnn! of the Fock space F(ℓ ⊗L2(X,σ)). 2 2 The proof of Theorem 4 is a straightforward generalization of the proof of The- orem 1 in [17], so we only outline it. First, we recall the classical unitary isomorphism between the usual Fock space over L2(R×X,ν ⊗σ) and L2(D′,µ): U : F(L2(R×X,ν ⊗σ)) → L2(D′,µ). 1 This isomorphism was broved by Itô [12] and extended in [16] to a general Lévy process on X. Note also that L2(R×X,ν ⊗σ) = L2(R,ν)⊗L2(X,σ). Next, we have the unitary operator U : L2(R,ν˜) → L2(R,ν) 2 defined by 1 (U f)(s) := f(s). 2 s Let U : ℓ → L2(R,ν˜) 3 2 be the Fourier transform in generalized joint eigenvectors of the Jacobi matrix J, see [2]. 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