Regular transformation groups based on Fourier- Gauss transforms

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1 ommunications on Stochastic Analysis Volume 9 Number Article Regular transformation groups based on Fourier- Gauss transforms Maximilian Bock Wolfgang Bock Follow this and additional works at: Part of the Analysis ommons, and the Other Mathematics ommons Recommended itation Bock, Maximilian and Bock, Wolfgang (015) "Regular transformation groups based on Fourier-Gauss transforms," ommunications on Stochastic Analysis: Vol. 9 : No., Article 3. DOI: /cosa Available at:

2 ommunications on Stochastic Analysis Vol. 9, No. (015) Serials Publications REGULAR TRANSFORMATION GROUPS BASED ON FOURIER-GAUSS TRANSFORMS MAXIMILIAN BOK AND WOLFGANG BOK Abstract. We discuss different representations of the white noise test functions (E) β, 0 β < 1 by introducing generalized Wick tensors. As an application we state the following generalization of the Mehler formula for the Ornstein-Uhlenbeck semigroup, see [7, p. 37]: Let a, b, 0 β < 1. Then a G + b N is the infinitesimal generator of the following regular differentiable one parameter transformation group { P a,b,t }t R GL((E) β): (i) if b 0 then for all φ (E) β, t R : P a,b,t (φ) = φ( (1 a b )(1 ebt ) x + e bt y) dµ(x) E (ii) if b = 0 then for all φ (E) β, t R : P a,0,t (φ) = φ( at x + y) dµ(x) E On an informal level the second case of the above theorem may be looked upon as a special case of the first one. Note that by the rules of l Hôpital we have lim (b a) 1 ebt = at. b 0 b 1. Introduction In finite dimensional analysis convolution and Fourier transform appear in many different applications. Since the Fourier transform is symmetric with respect to the dual pairing it extends naturally to generalized functions. The Fourier transform on (E) β resembles the finite dimensional Fourier transform in the sense that it is the adjoint of a continuous linear operator from the space of test functions into itself. This adjoint is a so called Fourier Gauss transform. In Gaussian Analysis, the Fourier-Gauss transform G a,b (φ) of φ (E) β is defined by G a,b (φ)(y) = E φ(ax + by) dµ(x), a, b, 0 β < 1 Received ; ommunicated by Hui-Hsiung Kuo. 010 Mathematics Subject lassification. Primary 60H40; Secondary 46F5. Key words and phrases. White noise theory, Gross Laplace operator, number operator, Fourier-Gauss transform, infinitesimal generator. convolution operator, generalized Wick tensors. 181

3 18 M. BOK AND W. BOK see e.g. [7], [11] and [10]. The Fourier-Gauss transform is in L((E) β, (E) β ) and the operator symbol is given by [ 1 ( Ĝ a,b (ξ, η) = exp a + b 1 ) ] ξ, ξ + b ξ, η, for all ξ, η E, see [10, Theorem 11.9, p ]. Hence G a,b = Γ(b Id) exp( 1 (a + b 1) G ), (1.1) where Id denotes the identity operator. onsidering this formula, we show that we can find a suitable representation of the white noise test functions, such that equation (1.1) will be just consisting of second quantization operators and is thus comparable to the approach in []. This will be done by introducing generalized Wick tensors. For m N, κ 0,m (E m ) sym we define generalized Wick tensors by : x n : κ0,m = n m k=0 n! (n mk)!k! ( 1 )k x (n mk) κ k 0,m. We show that for each test function φ in the space (E) β, with max(0,m ) m β < 1, we have a unique representation φ = : x n : κ0,m, φ n, with φ n (E n ) sym for all n N 0. Indeed we can rewrite the Fourier-Gauss transform G a,b as a suitable second quantization operator { Γ κ0,r (bid) }. Using generalized Wick tensors it is very natural to find a larger class of one parameter transformation groups, see As an application we deduce explicitly the regular one parameter groups corresponding to the infinitesimal generators a G + bn. In order to classify regularity we use and extend the result from [13, Theorem 5.6, p. 671]. There it is shown, that there exists a continuous homomorphism from (E) β to L((E) β, (E) β ). We show furthermore that this homomorphism is even an isomorphism from (E) β onto Im(). We use this theorem to prove easily the differentiability of some one parameter transformation groups. Moreover we want to show that these transformation groups regular. In particular we show that every differentiable one-parameter subgroup of GL(X), where X is a nuclear Fréchet space, has a regular generator, see 5.8. The investigation of regular transformation groups in this manuscript can be compared with [3].There the authors first construct a two-parameter transformation group G on the space of white noise test functions (E) in which the adjoints of Kuo s Fourier and Kuo s Fourier-Mehler transforms are included. They show that the group G is a two-dimensional complex Lie group whose infinitesimal generators are the Gross Laplacian G and the number operator N, and then find an explicit description of a differentiable one-parameter subgroup of G whose infinitesimal generator is a G + bn, which is identical with the one in this article.

4 FOURIER GAUSS TRANSFORMS 183. Preliminaries on White Noise Distribution Theory Starting point of the white noise distribution theory is the Gel fand triple E L (R, dt) E, where E is a nuclear real countably Hilbert space which is densely embedded in the Hilbert space of square integrable functions with respect to the Lebesgue measure L (R, dt) and E its topological dual space. Via the Bochner-Minlos theorem, see e.g. [1], we obtain the Gaussian measure µ on E by its Fourier transform exp(i x, ξ) dµ(x) = exp( 1 E ξ 0), ξ E where. 0 denotes the Hibertian norm on L (R, dt). The topology on E is induced by a positive self-adjoint operator A on the space of real-valued functions H = L (R, dt) with inf σ(a) > 1 and Hilbert-Schmidt inverse A 1. We set ρ := A 1 OP and δ := A 1 HS. Note that the complexification E are equipped with the norms ξ p := A p ξ 0 for p R. We denote H := L (R,, dt) furthermore } { } E,p := {ξ E ξ p < and Ep := ξ E ξ p <, for p R. Now we consider the following Gel fand triple of White Noise test and generalized functions. (E) β (L ) := L (E, µ) (E) β, 0 β < 1 By the Wiener-Itô chaos decomposition theorem, see e.g. [7, 1, 10] we have the following unitary isomorphism between (L ) and the Boson Fock space Γ(H ): (L ) Φ(x) = : x n :, f n (fn ) Φ Γ(H ), f n H ˆ n, (.1) where : x n : denotes the Wick ordering of x n and H ˆ n is the symmetric tensor product of order n of the complexification H of H. Moreover the (L ) - norm of Φ (L ) is given by Φ 0 = n! f n 0 The elements in (E) β are called white noise test functions, the elements in (E) β are called generalized white noise functions. We denote by.,. the canonical -bilinear form on (E) β (E) β. For each Φ (E) β there exists a unique sequence (F n ), F n (E ˆ n ) such that Φ, φ = n! F n, f n, (f n ) φ (E) β (.) Thus we have, see e.g. [7, 1, 10]: (E) β Φ (f n ),

5 184 M. BOK AND W. BOK ( if and only if for all p R we have Φ p,β := ) 1 <. Moreover for its dual space we obtain (n!) 1+β f n p (E) β Φ (F n ), (.3) ( ) 1 if and only if there exists a p R such that Φ p, β := (n!) 1 β F n p <. For p R we define and (E) p,β := (E) p, β := { } φ (E) β : φ p,β < { } φ (E) β : φ p, β <. It follows (E) β := proj lim(e) p,β p and (E) β = ind lim (E) p, β p Moreover (E) β is a nuclear (F)-space. In the following we use the abbreviation (E) := (E) 0. The exponential vector or Wick ordered exponential is defined by 1 Φ ξ (x) := : x n :, ξ n for ξ E and x E. (.4) n! For y E we use the same notation, which is only symbolic, and define Φ y (E) β by: (E) β ψ (f n ) n N0 : Φ y, ψ := y n, f n Since Φ ξ (E) β, for ξ E and 0 β < 1, we can define the so called S-transform of Ψ (E) β by S(Ψ)(ξ) = Φ ξ, Ψ, Moreover we call S(Ψ)(0) the generalized expectation of Ψ (E) β. The Wick product of Θ (E) β and Ψ (E) β is defined by Ψ Θ := S 1 (S(Ψ) S(Θ)) (E) β, see e.g. [7, 10, 1]. In the following we list some basic facts about integral kernel operators. Definition.1. Let l, m be nonnegative integers. For each κ l,m (E (l+m) ) the integral kernel operator Ξ l,m (κ l,m ) L((E) β, (E) β ) with kernel distribution κ l,m (see [1, Proposition 4.3.3, p. 8]) is defined by (n + m)! Ξ l,m (κ l,m )(φ) = : x (l+n) :, κ l,m m f n+m, φ (f n ) (E) β, n! (.5) where κ l,m m f n+m is the right contraction (see [1, Section 3.4, p. 53 ff]).

6 FOURIER GAUSS TRANSFORMS 185 Remark.. Note that : x (l+n) :, κ l,m m f n+m = : x (l+n) :, κ l,m ˆ m f n+m, because : x (l+n) : is symmetric and ˆ m means the right symmetric contraction. The correspondence κ l,m Ξ l,m (κ l,m ) is one-to-one if κ l,m (E (l+m) ) sym(l,m). Furthermore it is known (see [10, Section 10., p. 13 ff]): Ξ l,m (κ l,m ) L((E) β, (E) β ) κ l,m (E l ) (E m ) (.6) By a dual argument to [1, Theorem 4.3.9] it is known that an integral kernel operator Ξ l,m (κ l,m ) is extendable to on operator in L((E) β, (E) β ) if and only if κ l,m (E l ) (E m ). The operator symbol of Ξ L((E) β, (E) β ) is defined as ˆΞ(ξ, η) := Ξ(Φ ξ ), Φ η for ξ, η E. 3. onvergence of Generalized Functions Let X be a locally convex Hausdorff space and X its dual space with respect to the strong dual topology. Recall that L(X, X) is equipped with the topology of bounded convergence, namely the locally convex topology is defined by the semi-norms: T B,p := sup T ξ p, T L(X, X) (3.1) ξ B where B runs over the bounded subsets of X and p is an arbitrary semi-norm on X. The topology on L(X, X ) is defined in the same way. We state the following proposition: Proposition 3.1. Let X be a reflexive (F)-space and X its dual space with the strong dual topology. Then the mapping : L(X, X) L(X, X ), T T with T (ξ) := ξ T is a topological vector space isomorphism. Proof. For details see [6,.7 Satz, p. 84]. We present the main idea: Let M be a bounded subset of E and B be a bounded subset of X. Let T L(X, X). Since X is reflexive, the topological isomorphism follows by sup T (ξ) = sup pb ξ M ξ M,b B T ξ, b = sup T b pm (3.) b B The following theorem can be found in e.g. [9],[7, Theorem 4.41, p. 17 f] and [10, Theorem 8.6, p. 86f]. Theorem 3.. Let 0 β < 1. For all n N let Ψ n (E) β and F n = S(Ψ n ), where S denotes the S-transform. Then (Ψ n ) n N converges strongly in (E) β if and only if the following conditions are satisfied: (i) lim F n(ξ) exists for each ξ E n (ii) There exist nonnegative constants, K, p 0, independent of n, such that 1 β F n (ξ) exp(k ξ p ), n N, ξ E. (3.3)

7 186 M. BOK AND W. BOK As an example we consider the convergence of Wick exponentials. Lemma 3.3. Let ϕ (E) and ϕ (ϕ 0, ϕ 1,, ϕ m, 0, 0, 0, ) be its Wiener-Itô chaos decomposition. Then the expression exp 1 (ϕ) := n! ϕ n (3.4) converges in (E) max(0,m ) β, where 1 > β m. Proof. We show that the conditions from Theorem 3. are fullfilled. Without loss of let m 1 and ϕ m 0. Since ϕ (E) β there exists p > 0 such that ϕ p,β <. For n N 0 and ξ E let Ψ n := n k=0 1 k! ϕ k and F n (ξ) := n ondition (i) from Theorem 3. is fulfilled since n 1 F n (ξ) = k! (S(ϕ)(ξ))k exp(s(ϕ)(ξ)). k=0 It follows F n (ξ) exp( S(ϕ)(ξ) ). Since for r m we have S(ϕ)(ξ) ϕ ϕ m, ξ m ϕ 0 + m max 1 k m ( ϕ k p ) (1 + ξ r p ) k=0 1 k! ϕ k, Φ ξ. we obtain that condition (ii) from Theorem 3. is fullfilled for β 0 and r = 1 β. The following is an immediate consequence of Lemma 3.3 and comparisation of the S-transforms. orollary 3.4. Let ϕ, ψ (E) with corresponding chaos decompositions ϕ = (ϕ 0, ϕ 1,, ϕ m, 0, 0, 0,...) and ψ = (ψ 0, ψ 1,, ψ m, 0, 0, 0,...). Then the formula exp (ϕ + ψ) = exp (ϕ) exp (ψ) is valid in (E) β, for all 1 > β 1 m max(0, m ). 4. Wick Multiplication and onvolution Operator The Wick multiplication operator and its dual, often denoted as convolution operator are well known objects in white noise theory, see e.g. [13]. We give a proof, that (E) β with the Wick product may be considered as a commutative sub-algebra of L((E β ), (E β )). For a similar approach see also [13]. Proposition 4.1. (i) Let φ (E) β. The Wick multiplication operator M φ, defined by M φ (ψ) := φ ψ is a well defined operator in L((E) β, (E) β ). (ii) Let φ (E) β. The Wick multiplication operator M φ, defined by M φ (ψ) := φ ψ is a well defined continuous extension of M φ to an operator in L((E) β, (E) β ).

8 FOURIER GAUSS TRANSFORMS 187 Proof. For (ii), see [10, Theorem 8.1., p. 9]. To prove (i) we know by [10, Theorem 8.1., see Remark, p.9], that for any p 0 there exist suitable α > 0, c > 0, such that φ ψ p α, β Thus φ ( ) L((E) β, (E) β ). c φ p, β ψ p, β c φ p, β ψ p,β Definition 4.. The dual operator of the Wick operator M φ is called convolution operator and denoted by φ. The next proposition gives the Fock expansion of the Wick multiplication operator. Proposition 4.3. Let φ (E) β and φ (φ 0, φ 1, ). Then M φ, considered as M φ L((E) β, (E) β ), has the following Fock expansion: M φ = n N 0 Ξ n,0 (φ n ). Proof. Let ξ, η E. Then M φ Φ ξ, Φ η = S(φ Φ ξ )(η) = S(φ)(η) S(Φ ξ )(η) = S(φ)(η) e ξ,η By [10, (10.1) p. 145] the Taylor expansion of M φ is given by S(φ)(η) = φn, η n ξ 0 n N 0 completes the proof, compare also [1, Proposition 4.5.3, p.98-99]. Proposition 4.4. Let φ (E) β and φ (φ 0, φ 1, ). The convolution operator φ is in L((E) β, (E) β ) and has the Fock expansion φ = n N 0 Ξ 0,n (φ n ). Proof. By definition we have φ = M φ L((E)β, (E) β ). Moreover M φ (E) β = φ, since M φ is an extension of M φ. Then, by Proposition 4.3 we have ( ) φ = M φ = Ξ n,0 (φ n ) = Ξ 0,n (φ n ). n N 0 n N 0 The following statement is an immediate consequence from the definition of integral kernel operators. orollary 4.5. Let φ (E) β and φ (φ 0, φ 1, ). Then φ (Φ ξ ) = [S(φ)(ξ)] Φ ξ, ξ E :

9 188 M. BOK AND W. BOK Lemma 4.6. Let Ξ L((E) β, (E) β ) and Ξ = l,m=0 Ξ l,m (κ l,m ) be the corresponding unique representation as sum of integral kernel operators. Then: Ξ(Φ 0 ) = : x l :, κ l,0 Proof. onsider the equation (n + m)! Ξ l,m (κ l,m )(φ) = n! l=0 (4.1) : x (l+n) :, κ l,m ˆ m f n+m, φ (f n ) (E) β, then for Φ 0 (f n ) with f 0 = 1 and f n = 0 for all n 1: Ξ(Φ 0 ) = : x l :, κ l,0 ˆ 0 1 l=0 Theorem 4.7. Let 0 β < 1. The mapping : ((E) β, +,., ) to (L((E) β, (E) β ), +,., ) is an injective continuous vector algebra homomorphism. topological isomorphism from (E) β to Im(). φ φ (4.) In particular is a Proof. Linearity is given by the definition of integral kernel operators. In order to prove homomorphy, let ξ E, φ 1, φ (E) β. Then by orollary 4.5 φ1 φ (Φ ξ ) = S(φ 1 )(ξ) S(φ )(ξ) Φ ξ = S(φ 1 φ )(ξ) Φ ξ = φ1 φ (Φ ξ ) For injectivity let φ (E) β and φ = 0. Then φ = 0 and by 4.4 and 4.6 we obtain φ = φ (Φ 0 ) = 0. To prove continuity let φ (E) β and φ := : x n :, φ n be the corresponding chaos decomposition. Now let p 0. Moreover let r > 0, such that n N 0 ( 1 β ρ r < 1), where ρ := A 1 OP. hoose q > 0 such that ρ q. Then by [10, Theorem 10.5, p. 18] we have for all ϕ (E) β and m N 0 Ξ 0,m (φ m )(ϕ) p,β (m! m ) 1 β φ m (p+q+r) ϕ (p+q+r),β Now let K := ( m N 0 ( 1 β m N 0 Ξ 0,m (φ m )(ϕ) p,β m! 1 β φ m (p+q) ( 1 β ρ r ) m ϕ (p+q+r),β. ρ r ) m ) 1. Then by auchy-schwartz we obtain m N 0 m! 1 β φ m (p+q) ( 1 β ρ r ) m ϕ (p+q+r),β K φ (p+q), β ϕ (p+q+r),β

10 FOURIER GAUSS TRANSFORMS 189 Now let M (E) β be bounded. Then sup ϕ p+q+r,β const(m, p, q, r) < and ϕ M finally φ M,{p,β} K const(m, p, q, r) φ (p+q), β To prove the last claim, we use Proposition 3.1 and and represent 1 as composition of the two continuous mappings: 1 : φ (.) φ point evaluation φ (Φ 0 ) = φ. The following statement is a consequence from Lemma 3.3 and Theorem 4.7. orollary 4.8. Let φ (E) and φ (φ 0, φ 1,, φ m, 0, 0, 0, ) be it s chaos decomposition. Then the expression 1 exp( φ ) := n! ( φ) n (4.3) converges in L((E) β, (E) β ), where 1 > β max(0, m m ). 5. Regular One Parameter Groups Let X a nuclear (F)-space over and ( n ) n N be a family of Hilbertian seminorms, with n n+1 for all n N, topologizing X, see e.g. [1, Proposition 1.. and proposition 1.3.]. For x (X, n ) we define x n := sup x, x x n 1 By a Hahn-Banach argument, we have for all x X : x n = sup { x, x : x (X, n ) x n 1 } for all semi-norms n, n N. Definition 5.1. A family {Ω } R L(X) is called differentiable in 0 R, if Ω ϕ Ω 0 ϕ lim 0 0 converges in X for any ϕ X. In that case a linear operator Ω 0 from X into itself is defined by Ω Ω ϕ Ω 0 ϕ 0 ϕ := lim 0 0 {Ω } R is called differentiable if it is differentiable at each R. In the following we use the abbreviation: Ω := Ω 0 Proposition 5.. Let 0 R and {Ω } R L(X) be a family of operators which is differentiable in 0. Then Ω 0 is continuous, i.e. Ω 0 L(X). Moreover we have uniformly convergence on every compact (or equivalently, bounded) subset of X, i.e.: lim sup Ω ϕ Ω 0 ϕ 0 Ω ϕ K 0 ϕ 0 = 0 n for any n N and any compact (or bounded) subset K X.

11 190 M. BOK AND W. BOK Proof. First note that a subset of the nuclear space X is compact if and only if it is closed and bounded. The assertion follows by an application of the Banach- Steinhaus theorem. (For the uniform convergence on compact subsets see e.g. [14, 4.6 Theorem, p. 86]. ) Definition 5.3. A family {Ω } R L(X) is called regularily differentiable in 0 R, if there exists Ω 0 L(X) such that for any n N there exists m N such that lim sup Ω ϕ Ω 0 ϕ 0 Ω 0 ϕ 0 = 0. n ϕ m 1 {Ω } R is called regular if it is regularily differentiable at each R. Definition 5.4. Let {Ω } R be a family of operators in L(X) with 1, R : Ω 1 + = Ω 1 Ω, Ω 0 = Id Then, as easily seen, {Ω } R is a subgroup of GL(X) and is called a one-parameter subgroup of GL(X). In the following we collect facts about one-parameter subgroups, for details see e.g. [1, Section 5.]. Lemma 5.5. Let {Ω } R be a one-parameter subgroup of GL(X). (i) Then {Ω } R is differentiable at each R if and only if {Ω } R is differentiable at 0, i.e. there exists Ω L(X) such that lim Ω ϕ ϕ 0 Ω ϕ = 0. n for all ϕ X and n N. (ii) Let {Ω } R be a differentiable subgroup of GL(X). Then we have a) Ω is an element of L(X), further unique and is called the infinitesimal generator of the differentiable one parameter subgroup {Ω } R of GL(X). onversely a differentiable one parameter subgroup {Ω } R of GL(X) is uniquely defined by it s infinitesimal generator, see [1, Proposition 5.., p. 119]. If {Ω } R is regular the infinitesimal generator Ω is called a regular generator. b) {Ω } R is infinitely many differentiable at each R and d n n N : d n Ω = (Ω ) n Ω = Ω (Ω ) n c) R L(X), Ω is continuous. Note that L(X) is equipped with the topology of bounded convergence, compare [1, Section 5., Eq. (5.7), p. 119]. Lemma 5.6. Let T : R L(X) be a continuous mapping. If K is a compact subset of R, then T (K) is equicontinuous. Proof. T (K) is compact, hence bounded, especially pointwisely bounded, by the definition of the topology of bounded convergence. The theorem of Banach and Steinhaus then completes the proof.

12 FOURIER GAUSS TRANSFORMS 191 Proposition 5.7. Let {Ω } R be a differentiable one parameter subgroup of GL(X). (i) {Ω } R is regular if and only if for any n N there exists m N such that lim sup Ω ϕ ϕ 0 Ω ϕ = 0. n ϕ m 1 (ii) For any compact(or bounded) subset K X we have lim sup Ω ϕ ϕ 0 Ω ϕ = 0. n ϕ K (iii) Let 0 > 0. Then for any n N there exists m N and K(n, m) > 0, such that for all ϕ X sup Ω ϕ ϕ K(n, m) ϕ m n 0, 0 Proof. The statement (i) is easily verified, moreover (ii) is an immediate consequence of Proposition 5.. To prove statement (iii) we have by (ii) and Proposition 5.5 (ii) a), that the mapping { Ω Id R L(X),, 0 Ω, = 0 is continuous. Then the claim is obtained by Lemma 5.6. Theorem 5.8 (Regularity). Let {Ω } R be a differentiable one-parameter subgroup of GL(X). Then {Ω } R is regular. Proof. Let n N and ξ X, η (X, n ). We define for all t R : Then we have by Proposition 5.5 f(t) := η, Ω t ξ f (t) = η, Ω Ω t ξ f (t) = η, (Ω ) Ω t ξ Now let 0 > 0 be fixed. Then by Proposition 5.5 (ii) c) and Lemma 5.6, { (Ω ) Ω t } t 0 is equicontinuous. Thus there exists K = K( 0, n, m, Ω ) > 0 with m N 0 and such that max t 0 f (t) K ξ n+m η n Let R with 0. By Taylor expansion we have f() f(0) f (0) max t 0 f (t) and for 0 sup sup ξ n+m 1 η n 1 η, Ω ξ η, ξ K ξ n+m η n η, Ω ξ K.

13 19 M. BOK AND W. BOK Hence such that sup ξ n+m 1 lim 0 ξ n+m 1 Ω ξ ξ sup Ω ξ ξ Ω ξ n K Ω ξ = 0. n Proposition 5.9. Let {S } R be a differentiable one parameter subgroup of GL(X) and {T } R be a family of operators in L(X) which is differentiable in zero with lim T ϕ = ϕ for all ϕ X. Then 0 d d =0 S T = S + T If {T } R is regularily differentiable in zero, then is also regularily differentiable in zero. {S T } R Proof. Let n N, ϕ X. Then {S 1} is equicontinuous by Proposition 5.5, (ii) c). Then there exist a semi-norm m, with m N and m n, and a constant K > 0 such that S (ϕ) n K ϕ m for all ϕ X and R with 1. Now let ϕ X be arbitrarily chosen. For R with 1 we have: S T (ϕ) ϕ S ϕ T ϕ = S (T (ϕ) ϕ) + S (ϕ) ϕ n S (T (ϕ) ϕ) T ϕ + S (ϕ) ϕ S ϕ n n K (T (ϕ) ϕ) S T ϕ + S (ϕ) ϕ S ϕ m n K (T (ϕ) ϕ) T ϕ + K S (T ϕ) (T ϕ) m + m S ϕ T ϕ S (ϕ) ϕ where last inequalilty follows by Theorem 5.8 and the continuity of T. We use the following notation, due to [1, Eq. (4.66), p. 106] Definition γ n (T ) := n 1 γ 0 (T ) := 0 k=0 Id k T Id (n 1 k), n 1 n S ϕ, n Now let T L(E ). We recall the definition of the second quantization operator of T, denoted by Γ(T ) and dγ(t ), the differential second quantization operator. Suppose ϕ (E) β is given as ϕ(x) = : x n :, f n, x E, f n (E n ) sym

14 FOURIER GAUSS TRANSFORMS 193 as usual. We put and Γ(T )ϕ(x) := dγ(t )ϕ(x) := : x n :, T n f n : x n :, γ n (T )f n We have Γ(T ), dγ(t ) L((E) β ), for all 0 β < 1. Lemma Let {Ω } R be a differentiable one parameter subgroup of GL(X) with infinitesimal generaor Ω. Then, for all n N, { Ω n } is a differentiable R one parameter subgroup of GL(X n ) with d d =0 Ω n = γ n (Ω ) Proof. We show the case n =. The general case is similar. It holds Now apply Proposition 5.9 Ω Ω = (Ω Id) (Id Ω ) Proposition 5.1. Let 0 β < 1. Further let {Ω } R be a differentiable one parameter subgroup of GL(E ) with infinitesimal generator Ω. Then Γ(Ω ) R is a differentiable one-parameter subgroup of GL((E) β ) with infinitesimal generator dγ(ω ). Proof. Note that for 0 β < 1, the sequence (α(n)) n N0 with α(n) := n! β fullfills the conditions of [5, Theorem 4., p.696], where a detailed proof, based on the characterization theorem, is given. On the other hand the expected result is easily seen by Lemma For a similar statement like Proposition 5.1, see [1, 5.4.5, p ]. Proposition Let {Ω } R be a differentiable one parameter subgroup of GL(E ). Furthermore let r N, 1 > β max(0,r ) r and κ 0,r (E r ) sym. Then { exp(ξ 0,r ((Ω r ) κ 0,r )) } R GL((E) β) is an in zero differentiable family of operators with d d =0 exp(ξ 0,r ((Ω r ) κ 0,r )) = Ξ 0,r ((γ r (Ω )) κ 0,r ) exp(ξ 0,r ((κ 0,r )). Proof. First, because R is a metric space, it is enough to consider sequential convergence, i.e. the limit process for any arbitrary sequence ( n ) n N in R with lim n = 0. onsider the sequence { exp : x r :, (Ω r n n ) κ 0,r. Then the }n N limit process will be transferred to L((E) β, ((E) β )) by continuity, using Theorem 4.7. Now let ( n ) n N be an arbitrary sequence in R with lim n = 0 and n 0 n for all n N. Further define φ n := exp ( : x r :, (Ω r n ) κ 0,r ) exp ( : x r :, κ 0,r ) n

15 194 M. BOK AND W. BOK for all n N. Then, by Lemma 3.3, we have φ n (E) β for all n N. We verify the conditions of Theorem 3.. Let ξ E. Then for all n N S(φ n )(ξ) = exp( κ 0,r, (Ω r n )ξ r ) exp( κ 0,r, ξ r ) n First note that for all n N the S-transform S(φ n ) is entire holomorphic. By Lemma 5.11 and Proposition 5.5 we obtain { Ω r subgroup of (E r ) sym with d d Ω r } = γ r (Ω )Ω r. R as a regular one-parameter Hence the function κ 0,r, (Ω r )ξ r is infinitely often differentiable on R and the same holds for exp( κ 0,r, (Ω r )ξ r ) as composition of two infinitely many differentiable functions with d d exp( κ 0,r, (Ω r )ξ r ) = κ 0,r, γ r (Ω )Ω r ξ r exp( κ 0,r, (Ω r )ξ r ) Hence lim (S(φ n)(ξ)) exists and we have: n lim S(φ n)(ξ) = κ 0,r, γ r (Ω )ξ r exp( κ 0,r, ξ r ) n by the chain rule and Proposition 5.1. For the growth estimate let p 0. Without loss of generality let n < 1 for all n N. Then, since {Ω } 1 and {Ω Ω } 1 are compact by 5.6, there exists a q 0 such that, R, 1 we have Ω (ξ) p ξ p+q and Ω Ω (ξ) p ξ p+q. Then, by the mean value theorem and by Proposition 5.5 (ii) b), it follows for each n N: S(φ n )(ξ) sup 1 κ 0,r, γ r (Ω )Ω r ξ r exp( κ 0,r, (Ω r )ξ r ) κ 0,r p r ξ r p+q exp( κ 0,r p ξ r p+q )) κ 0,r p r(r!) exp((1 + κ 0,r p ) ξ r p+q )) κ 0,r p r(r!) exp((1 + κ 0,r p ) (1 + ξ 1 β p+q )), for all 1 > β max(0,r ) r. The claim is a consequence of Theorem 4.7. The same idea as in the proof of Proposition 5.13 combined with Theorem 5.8 leads to the following result. Proposition Let r N, 1 > β 1 r max(0, r ). Further let φ (E) β with φ (φ 0,, φ r, 0, 0, 0, ). Define Ξ := exp( φ ) for R. Then {Ξ : R} is a regular one parameter subgroup of GL((E) β ) with infinitesimal generator φ. Example Let y E. As simple example consider the operator Ξ 0,1 (y). Recall that D y = Ξ 0,1 (y) and the translation operator T y = exp(d y ). Let z. From z G = Ξ 0, (zτ), where τ is the trace operator, we conclude that exp(z G ) L((E) β, (E) β ), for all 0 β < 1.

16 FOURIER GAUSS TRANSFORMS Generalized Wick Tensors and an Application Our goal in this section is to rewrite Fourier-Gauss transforms as second quantization operators. This will be accomplished by suitable basis-transformations which will be explicitly calculated. For this purpose we introduce generalized Wick tensors. In this context Fourier-Gauss transforms appear as second quantization operators of the form { Γ κ0,r (Ω ) }. As an application we deduce explicitly the regular one parameter groups corresponding to the infinitesimal generators a G + bn. First we repeat some definitions. Let a, b, 0 β < 1. The Fourier-Gauss transform G a,b (φ) of φ (E) β is defined to be the function G a,b (φ)(y) = φ(ax + by) dµ(x) E The Fourier-Gauss transform is in L((E) β, (E) β ) and the operator symbol is given by [ 1 ( Ĝ a,b (ξ, η) = exp a + b 1 ) ] ξ, ξ + b ξ, η, for all ξ, η E see e.g. [10, Theorem 11.9, p ]. Hence G a,b = Γ(b Id) exp( 1 (a + b 1) G ) By [10, Lemma 11., p. 163] the operator symbol of the Fourier transform is given by onsequently F(ξ, η) = exp( i ξ, η 1 η, η), for all ξ, η E. F = exp( 1 G) Γ( i Id) Moreover the operator symbol of the Fourier-Mehler transform is given by F (ξ, η) = exp(e i ξ, η + i ei sin η, η), for all ξ, η E, R, see e.g. [10, 11., p. 180]. Thus the Fourier-Mehler transform is given by the formula [ F = exp( i ei sin G )] Γ(e i Id). In the following let G denote the adjoint of the Fourier-Mehler transform F. Finally by [1, Proposition 4.6.9, p. 105] the operator symbol of the scaling operator is given by Ŝ λ (ξ, η) = exp ( (λ 1) ξ, ξ / + λ ξ, η ), for all ξ, η E, λ. and consequently ( λ 1 S λ = Γ(λ Id) exp G )

17 196 M. BOK AND W. BOK Definition 6.1. Let m N and κ 0,m ((E ) m ) sym. For x E we define the renormalized tensor power : x n : κ0,m as follows: : x n : κ0,m = n m k=0 ( n! (n mk)!k! 1 ) k x (n mk) (κ 0,m) k For the usual tensor power x n, x E it holds the following relation:, see [1, orollary..4, p. 5] x n = n k=0 n! (n k)!k! ( ) k 1 : x (n k) : τ (τ) k As a simple example we have for x E the relation : x n : = : x n : τ. More usually is the abbreviation : x n def : σ = : x n : σ τ. Note that κ 0,m = 0 is permitted, e.g. : x n : 0 = x n. But we don t permit m = 0. There exists Θ in L((E), (E)) defined by Θ(exp(., ξ)) := Φ ξ, see e.g. [10, Theorem 6.]. Since Φ ξ dµ = 1 we call Θ the renormalization operator. E Proposition 6.. (i) Θ = exp( 1 G) (ii) For all f m (E m ) sym we have: Θ( x m, f m ) = : x m :, f m Proof. By orollary 4.5 we have for all ξ E : exp( 1 G)Φ ξ = exp( 1 ξ, ξ)φ ξ = exp( 1 ξ, ξ) exp( 1 = e.,ξ ξ, ξ)e.,ξ By Proposition 5.14 Θ is invertible and the first statement is proved. To proof the second, let m N 0. Note that for m < n we have further Θ 1 = exp( 1 G) = Ξ 0,n (τ n )( : x m :, f m ) = 0, 1 n n! Ξ 0,n(τ n ). Let m n and δ i,j be the Kronecker symbol. Then by [1, Proposition 4.3.3, Eq. (4.3), p. 8] it holds: Ξ 0,n (τ n )( : x m :, f m ) = k=0 (k + n)! ( : x k :, τ n n δ k+n,m f m ) k! m! = : x m n :, τ n n f m (m n)! m! = : x m n : τ n, f m (m n)! m! = : x m n : ˆ τ ˆ n, f m (m n)!

18 FOURIER GAUSS TRANSFORMS 197 where the last equation is due to the symmetricity of f m. Then exp( 1 G)( m : x m m! :, f m ) = (m n)!n! n : x m n : ˆ τ ˆ n, f m compare also[1, orollary..4]. = x m, f m, Note that e,ξ (E) since Φ ξ (E). orollary 6.3. Let ξ E. Then the series e,ξ = (E). Proof. Let ξ E. Since converges in (E) also converges in (E). Φ ξ = exp( 1 G)Φ ξ = 1 : x n :, ξ n n! 1 x n, ξ n n! The following proposition generalizes Proposition n!., ξn converges in Proposition 6.4. Let r N and κ 0,r ((E ) r ) sym. For all f m (E m ) sym we have: exp( 1 Ξ 0,r(κ 0,r ))( x m, f m ) = : x m : κ0,r, f m Proof. Let m N 0. Note that for m < rn we have further exp( 1 Ξ 0,r(κ 0,r )) = Ξ 0,rn (κ n 0,r )( : x m :, f m ) = 0, 1 n! ( 1 )n Ξ 0,rn (κ n 0,r ). Let m rn and δ i,j be the Kronecker symbol. Then by Theorem 4.7 and Proposition 6., using [1, Proposition 4.3.3, Eq. (4.3), p. 8], we have: Ξ 0,rn (κ n 0,r )( x m 1, f m ) = exp( G) Ξ 0,rn (κ n 0,r )( : x m :, f m ) = exp( 1 (k + rn)! G) ( : x k :, κ n 0,r k! rn δ k+rn,m f m ) = = m! (m rn)! m! (m rn)! k=0 x m rn, κ n 0,r rn f m x m rn κ n 0,r, f m! m = (m rn)! where the last equation follows because f m is symmetric. x m rn ˆ κ ˆ n 0,r, f m

19 198 M. BOK AND W. BOK Then by the above definition we have: exp( 1 Ξ 0,r(κ 0,r ))( m r x m m!, f m ) = (m rn)!n! ( 1 )n x m rn ˆ κ ˆ n 0,r, f m = : x m : κ0,r, f m Notation 6.5. In the following we use 0 τ in order to express that we consider 0 (E ). Theorem 6.6 (Representation theorem). Let r N, 1 > β max(0,r ) r and κ 0,r ((E ) r ) sym. Then each (φ n ) n N0 φ (E) β has a unique decomposition with (ψ n ) κ0,r φ(x) = : x n : κ0,r, (ψ n ) κ0,r, for all x E (E n ) sym, which we denote as κ 0,r - representation of φ. Proof. First note that φ has a unique Wiener-Itô chaos decomposition (φ n ) n N0, see e.g. [1, Theorem 3.1.5]. Then the existence and uniqueness of the above representation follows by the bijectivity of exp( 1 G) exp( 1 Ξ 0,r(κ 0,r )). Recall orollary 4.8 and the following chain of mappings: : x n : κ0,r, φ n exp( 1 Ξ 0,r(κ 0,r )) x n, φ n exp( 1 G) : x n :, φ n Remark 6.7. The S-transform of φ (E) is a restriction of the 0τ - representation of exp( 1 G)φ from E to E. Note that we do not claim that the above decomposition is an orthogonal decomposition with respect to the measure µ, like the chaos decomposition. We claim only the uniqueness. Definition 6.8. Let r N, 1 > β max(0,r ) r, κ 0,r (E r ) sym. For T L((E) β, (E) β ), we define: T κ0,r := exp( 1 ( G Ξ 0,r (κ 0,r )) T exp( 1 ( G Ξ 0,r (κ 0,r ))) T κ0,r is called the renormalization of T corresponding to κ 0,r. Obviously T κ0,r L((E) β, (E) β ). For Ω L((E) β ) we abbreviate Γ κ0,r (Ω) := (Γ(Ω)) κ0,r, dγ κ0,r (Ω) := (dγ(ω)) κ0,r. Remark 6.9. It is clear, that T κ0,r acts formally on white noise test functions in κ 0,r - representation like T on white noise test functions in the standard representation as Boson Fock space. We precise this statement by the following proposition.

20 FOURIER GAUSS TRANSFORMS 199 Proposition Let r N, 1 > β max(0,r ) r, κ 0,r (E r ) sym. For T L((E) β, (E) β ) and φ (E) β with φ = : x n :, f n, we use the notation T (φ) = : x n :, (f n ) T. Then it follows: T κ0,r ( : x n : κ0,r, f n = : x n : κ0,r, (f n ) T Proof. The expression : x n : κ0,r, f n is well defined because : x n : κ0,r, f n = exp( 1 Ξ 0,r(κ 0,r )) exp( 1 G)( By orollary 3.4 and Theorem 4.7 we have : x n :, f n ). exp( 1 ( G Ξ 0,r (κ 0,r ))) = exp( 1 G) exp( 1 Ξ 0,r(κ 0,r )). Then using Proposition 6. and Proposition 6.4, the claim follows with the same idea as in the proof of Theorem 6.6. The following formula is suitable for the calculation of renormalized second quantization operators. Recall that for all T L(E, E ), we have Γ(T ) L((E) β, (E) β ), where 0 β < 1. orollary Let r N, 1 > β max(0,r ) r, κ 0,r (E r ) sym. Then for all T L(E, E ) we have Γ κ0,r (T ) = Γ(T ) exp( 1 Ξ 0,((T Id ) (τ))) Proof. On the one hand we calculate exp( 1 Ξ 0,r((T r Id r ) (κ 0,r ))). Γ(T ) exp( 1 Ξ 0,((T Id ) (τ))) exp( 1 Ξ 0,r((T r Id r ) (κ 0,r )))Φ ξ = (exp( 1 τ, (T 1)ξ )) (exp( 1 (κ0,r ), (T r 1)ξ r ))Φ T ξ On the other hand by Definition 6.8 Γ κ0,r (T )(Φ ξ ) = exp( 1 ( G Ξ 0,r (κ 0,r ))) Γ(T ) exp( 1 ( G Ξ 0,r (κ 0,r )))(Φ ξ ) = exp( 1 ( ξ, ξ > < κ 0,r, ξ r )) exp( 1 ( G Ξ 0,r (κ 0,r )))(Φ T ξ ) = exp( 1 ( ξ, ξ > < κ 0,r, ξ r )) exp( 1 ( T ξ, T ξ κ 0,r, (T ξ) r ))Φ T ξ = (exp( 1 τ, (T 1)ξ )) (exp( 1 (κ0,r ), (T r 1)ξ r ))Φ T ξ

21 00 M. BOK AND W. BOK We consider some concrete examples to orollary Example 6.1. Let b, R and κ 0, (E ) sym. Then (i) (ii) (iii) Γ κ0, (b Id) = Γ(b Id) exp( 1 (b 1)Ξ 0, (τ κ 0, )) Γ κ0, (e i Id) = Γ(e i Id) exp ( i e i sin Ξ 0, (τ κ 0, ) ) ( ) i Γ 1 τ (e i Id) = Γ(e i Id) exp ei sin G By [1, Lemma 5.6.1, p.140] we conclude that Γ 1 τ (e i Id) is the adjoint operator of the Fourier-Mehler transform F. The next example gives conditions under which the Fourier-Gauss transforms are renormalized second quantization operators. Example Let a, b and b / { 1, 1}. hoose σ with σ = Γ σ τ (b Id) = G a,b Which follows immediately by Example 6.1(i) with κ 0, = σ τ. a 1 b. Then Remark In the special case a = 0, we have for the scaling operator S b := G 0,b = Γ 0τ (b Id), i.e. S b ( x n, f n ) = x n, b n f n for all n N 0, f n (E n ) sym. With a, b, b / { 1, 1} and σ = a 1 b, we have Γ σ κ 0, (b Id) = Γ(b Id) exp( 1 [ a Ξ 0, (κ 0, ) + (b ] 1) G ) Theorem Let {T } R be a differentiable one parameter subgroup of GL(E ) with infinitesimal generator T. Further let r N, 1 > β max(0,r ) r and κ 0,r (E r ) sym. Then { Γ κ0,r (T ) } is a regular one parameter subgroup of R GL((E) β ) with dγ κ0,r (T ) = d d =0 Γ κ0,r (T ) = dγ(t ) + 1 Ξ 0,(γ (T ) τ) 1 Ξ 0,r(γ r (T ) κ 0,r ) Proof. By orollary 6.11 we have Γ κ0,r (T ) = Γ(T ) exp( 1 Ξ 0,((T Id ) (τ))) exp( 1 Ξ 0,r((T r Id r ) (κ 0,r ))). First note that (G Γ(T ) G 1 ) R is obviously a regular one-parameter subgroup of GL((E) β ) for all G GL((E) β ). onsequently (Γ 0τ (T )) R, with Γ 0τ (T ) = Γ(T ) exp( 1 Ξ 0,((T Id ) (τ))), is a regular one-parameter subgroup of GL((E) β ). Because

22 Γ κ0,r (T ) = FOURIER GAUSS TRANSFORMS 01 [ Γ(T ) exp( 1 ] Ξ 0,((T Id ) (τ))) exp( 1 Ξ 0,r((T r Id r ) (κ 0,r ))), the claim follows by Proposition 5.9, then Proposition 5.1 and Proposition orollary Let a, b and b 0. Then bn (1 a b )τ = d d =0 Γ (1 a b )τ (e b Id) = a G + bn. Proof. By [1, Proposition , p.107] we have N := Ξ 1,1 (τ) = dγ(id). Because bid is the infinitesimal generator of { e b Id }, it follows by Proposition 5.1 that R d d =0 (Γ(e b Id)) = dγ(b Id) = Ξ 1,1 (bτ) = bn, for b. On the other hand, in the general case r N, 1 > β max(0,r ) r and κ 0,r ) sym, it holds from Theorem 6.15, with T = Id, that (E r N κ0,r = d d =0 Γ κ0,r (e Id) = N + G r Ξ 0,r(κ 0,r ). (6.1) Multiplicating both sides with b, the claim is immediate with κ 0,r = (1 a b )τ. Remark Let b 0. The regular one-parameter subgroup { } Γ (1 a b )τ (e b Id) GL((E) β ), 0 β < 1 R can, Example 6.13, be identified as the Fourier-Gauss transforms { G x,e b}, where x = (1 a b )(1 eb ). { } The case b = 0 is solved by We get G a,1 as solution. Note, that the special choice of x from x has no influence, because { } G x,e b only depends on x and e b. We summarize this discussion using the definition of the Fourier-Gauss transform in [10, Definition 11.4, p. 164]. The following theorem is a generalization of the Mehler formula for the Ornstein-Uhlenbeck semigroup, see e.g. [7, p. 37] Theorem Let a, b, 0 β < 1. Then a G + b N is the infinitesimal generator of the following regular transformation group {P a,b,t } t R GL((E) β ): (i) if b 0 then for all φ (E) β, t R: P a,b,t (φ) = φ( (1 a b )(1 ebt ) x + e bt y) dµ(x) E (ii) if b = 0 then φ (E) β, t R: P a,0,t (φ) = φ( at x + y) dµ(x) E

23 0 M. BOK AND W. BOK On an informal level the second case of the above theorem may be considered as a special case of the first one. Note that by the rules of l Hôpital we have 1 ebt (b a) = at. lim b 0 b 7. Summary and Bibligraphical Notes The investigation of regular transformation groups in this manuscript can be compared with [3], where t a two-parameter transformation group G on the space of white noise test functions (E) was conctructed, which includes the adjoints of Kuo s Fourier and Kuo s Fourier-Mehler transforms. Their description of a differentiable one-parameter subgroup of G whose infinitesimal generator is a G + bn is identical with our findings. However using the unique representation of white noise test functions via generalized Wick tensors and corresponding renormalized operators we have presented a different way which leads to a more general result for the spaces (E) β. Using convolution operators and the fact that differentiable one parameter transformation groups on nuclear Fréchet spaces are always regular, convergence problems could be solved. Acknowledgment. The authors would like to thank to Prof. H.-H. Kuo for his encouragement to publish this manuscript. W. Bock would like to thank for the financial support from the FT project PTD/MAT-STA/184/01. We thank the referees for their helpful suggestions to improve the presentation of this article. References 1. Berezansky, Y. M. and Kondratiev, Y. G. (1995): Spectral Methods in Infinite-dimensional Analysis. Vol.. Dordrecht: Kluwer Academic Publishers. Translated from the 1988 Russian original by Malyshev P. V. and Malyshev D. V. and revised by the authors.. Bock,W.: Generalized Scaling Operators in White Noise Analysis and Applications to Hamiltonian Path Integrals with Quadratic Action, in Stochastic and Infinite Dimensional Analysis, Springer, hung, D. M. and Ji, U..: Transformation Groups on White Noise Functionals and Their Applications, Applied Mathematics & Optimization, 1998, Volume 37, Number, Seiten hung, D. M., Ji, U.., and Obata, N.: Transformations on White Noise functions associated with second order differential operators of diagonal type, Nagoya Math. J., 149 (1998), hung,.-h., hung, D. M., and Ji, U..: One-parameter groups and cosine families of operators on white noise functions, Bull. Korean. Math. Soc. 37 (000), No. 5, Floret, K. and Wloka, J.: Einführung in die Theorie der Lokalkonvexen Räume, Lecture Notes in Mathematics 1968, (56) Walter Springer-Verlag, Berlin Heidelberg New York. 7. Hida, T., Kuo, H.-H., Potthoff, J., and Streit L.: White Noise: An Infinite-dimensional alculus, Kluwer Academic Publishers, Dordrecht, Hida, T., Obata, N., and Saitô, K.: Infinite dimensional rotations and Laplacians in terms of white noise calculus, Nagoya Math. J. 18 (199), Kondratiev, Yu. G. Leukert, P., Potthoff, J., Streit, L., and Westerkamp, W.: Generalized functionals in Gaussian spaces: The characterization theorem revisited. J. Funct. Anal. 141 (1996) Nr Kuo, H.-H.: White Noise Distribution Theory, R Press, Boca Raton, Lee, Y.-J.: Integral transforms of analytic functions on abstract Wiener spaces, J. Funct. Anal. 47 (1983)

24 FOURIER GAUSS TRANSFORMS Obata, N.: White Noise Analysis and Fock Space, Lecture Notes in Mathematics, Vol. 1577, Springer-Verlag, Berlin, Obata, N. and Ouerdiane, H.: A note on convolution operators in white noise calculus, IDAQP, 14, No. 4 (011) , World Scientific Publishing ompany, DOI: /S Schaefer, H. H. and Wolff, M. P.: Topological Vector Spaces, Springer-Verlag New York Berlin Heidelberg, second edition 1999, ISBN SPIN Maximilian Bock: Saarstahl AG, Informatik, Bismarckstr , Völklingen, Germany address: maximilianbock61@gmail.com Wolfgang Bock: Technomathematics Group,Department of Mathematics, University of Kaiserslautern, Kaiserslautern, Germany address: bock@mathematik.uni-kl.de