Unitary Processes with Independent Increments and Representations of

45(2009), 745–785

Unitary Processes with Independent Increments and Representations of

Hilbert Tensor Algebras

By

LingarajSahu^∗, MichaelSch¨urmann^∗∗and Kalyan B. Sinha^∗∗∗

Abstract

The aim of this article is to characterize unitary increment process by a quantum stochastic integral representation on symmetric Fock space. Under certain assump- tions we have proved its unitary equivalence to a Hudson-Parthasarathy flow.

§1. Introduction

In the framework of the theory of quantum stochastic calculus developed by pioneering work of Hudson and Parthasarathy [6], quantum stochastic dif- ferential equations (qsde) of the form

(1.1) dV_t=

μ,ν≥0

V_tL^μ_νΛ^ν_μ(dt), V₀= 1_h⊗Γ,

(where the coefficients L^μ_ν : μ, ν ≥ 0 are operators in the initial Hilbert spaceh and Λ^ν_μ are fundamental processes in the symmetric Fock space Γ = Γ_sym(L²(R₊,k)) with respect to a fixed orthonormal basis (in short ‘ONB’)

Communicated by Y. Takahashi. Received January 7, 2008. Revised December 2, 2008.

2000 Mathematics Subject Classification(s): 81S25, 47D03, 60G51.

∗Stat-Math Unit, Indian Statistical Institute, Bangalore Centre, 8^thMile, Mysore Road, Bangalore-59, India.

e-mail: lingaraj@gmail.com

∗∗Institut f¨ur Mathematik und Informatik, F.-L.-Jahn-Strasse 15a, D-17487 Greifswald, Germany.

e-mail: schurman@uni-greifswald.de

∗∗∗Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore-64, and Department of Mathematics, Indian Institute of Science, Bangalore-12, India.

e-mail: kbs jaya@yahoo.co.in

c 2009 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.

{E_j :j ≥1} of the noise Hilbert space k) have been formulated. Conditions for existence and uniqueness of a solution {V_t} are studied by Hudson and Parthasarathy and many other authors. In particular when the coefficients L^μ_ν :μ, ν ≥0 are bounded operators satisfying some conditions it is observed that the solution{V_t:t≥0}is a unitary process.

In [4], using the integral representation of regular quantum martingales in symmetric Fock space [17], the authors show that any covariant Fock adapted unitary evolution {V_s,t : 0≤s ≤t < ∞} (with norm-continuous expectation semigroup) satisfies a quantum stochastic differential equation (1.1) with con- stant coefficients L^μ_ν ∈ B(h).For situations where the expectation semigroup is not norm continuous, the characterization problem is discussed in [5, 1].

In [10, 11], by extended semigroup methods, Lindsay and Wills have studied such problems for Fock adapted contractive operator cocycles and completely positive cocyles.

In this article we are interested in the characterization of unitary evolutions with stationary and independent increments on h⊗ H, where h and H are separable Hilbert spaces. In [18, 19], by a co-algebraic treatment, the second author has proved that any weakly continuous unitary stationary independent increment process onh⊗ H,hfinite dimensional, is unitarily equivalent to a Hudson-Parthasarathy flow with constant operator coefficients; see also [8, 9].

In this present paper we treat the case of a unitary stationary independent increment process onh⊗ H,hnot necessarily finite dimensional, with norm- continuous expectation semigroup. By a GNS type construction we are able to get the noise space k and the bounded operator coefficients L^μ_ν such that the Hudson-Parthasarathy flow equation (1.1) admits a unique unitary solution and is unitarily equivalent to the unitary process we started with.

The article is organized as follows: Section 2 is meant for recalling some preliminary ideas and fixing some notations on linear operators on Hilbert spaces and quantum stochastic flows on Fock space. In the next Section an algebra structure is given on tensor product of Hilbert space which we are calling as Hilbert tensor algebra. The unitary processes with stationary and independent increments are described in Section 4 and filtration property of these processes is seen in Section 5. In Section 6 various semigroups associated with above mentioned unitary processes are studied and using them a Hilbert space, called noise space and structure maps are constructed from the Hilbert tensor algebra in Section 7. Associated Hudson-Parthasarathy flow is studied in Section 8 and its minimality is discussed in Section 9. In the last Section unitary equivalence to Hudson-Parthasarathy flow is established.

§2. Notation and Preliminaries

We assume that all the Hilbert spaces appearing in this article are complex separable with inner product anti-linear in the first variable. For any Hilbert spacesH,K B(H,K) andB₁(H) denote the Banach space of bounded linear operators fromHtoKand trace class operators onHrespectively. For a linear (not necessarily bounded) mapT we write its domain asD(T).We denote the trace onB1(H) by T r_H or simply T r.The von Neumann algebra of bounded linear operators onHis denoted byB(H).The Banach spaceB1(H,K)≡ {ρ∈ B(H,K) :|ρ|:=√

ρ^∗ρ∈ B₁(H)}with norm (Ref. Page no. 47 in [2]) ρ ₁= |ρ| _B1(H)= sup{

k,l

|φ_k, ρψ_l|:{φ_k},{ψ_l}are ON B of KandHresp.}

is the predual of B(K,H). For an element x ∈ B(K,H), B₁(H,K) ρ → T r_H(xρ) defines an element of the dual Banach space B₁(H,K)^∗.For a linear map T on the Banach space B₁(H,K) the adjointT^∗ on the dual B(K,H) is given byT r_H(T^∗(x)ρ) :=T r_H(xT(ρ)), ∀x∈ B(K,H), ρ∈ B1(H,K).

For any ξ∈ H ⊗ K, h∈ Hthe map

K k→ ξ, h⊗k

defines a bounded linear functional on K and thus by Riesz’s theorem there exists a unique vectorξ, hinK such that

(2.1) ξ, h, k=ξ, h⊗k,∀k∈ K.

In other wordsξ, h=F_h^∗ξwhereF_h∈ B(K,H ⊗K) is given byF_hk=h⊗k.

LethandHbe two Hilbert spaces with some orthonormal bases{e_j :j≥ 1} and{ζ_n:n≥1} respectively. For A∈ B(h⊗ H) and u, v∈hwe define a linear operatorA(u, v)∈ B(H) by

ξ₁, A(u, v)ξ₂=u⊗ξ₁, A v⊗ξ₂, ∀ξ₁, ξ₂∈ H and read off the following properties:

Lemma 2.1. Let A, B∈ B(h⊗ H)then for any u, v, u_i andv_i, i= 1,2 inh

(i) A(u, v)∈ B(H)with A(u, v) ≤ A u v andA(u, v)^∗=A^∗(v, u).

(ii) h×h→A(·,·)is1−1, i.e. ifA(u, v) =B(u, v), ∀u, v∈hthenA=B.

(iii) A(u₁, v₁)B(u₂, v₂) = [A(|v₁>< u₂| ⊗1_H)B](u₁, v₂) (iv) AB(u, v) =

j≥1A(u, e_j)B(e_j, v) (strongly) (v) 0≤A(u, v)^∗A(u, v)≤ u ²A^∗A(v, v)

(vi) A(u, v)ξ₁, B(p, w)ξ₂=

n≥1p⊗ζ_n,[B(|w >< v|⊗|ξ₂>< ξ₁|)A^∗u⊗ζ_n

=v⊗ξ₁, [A^∗(|u >< p| ⊗1_H)Bw⊗ξ₂. Proof. We are omitting the proof of (i), (ii).

(iii) For anyξ, ζ∈ Hwe have

ξ, A(u₁, v₁)B(u₂, v₂)ζ=u₁⊗ξ, Av₁⊗B(u₂, v₂)ζ

=A^∗u₁⊗ξ, v₁⊗B(u₂, v₂)ζ

=

n≥1

A^∗u₁⊗ξ, v₁⊗ζ_nζ_n, B(u₂, v₂)ζ

=

n≥1

A^∗u₁⊗ξ, v₁⊗ζ_nu₂⊗ζ_n, Bv₂⊗ζ

=

n≥1

A^∗u₁⊗ξ,(|v₁>< u₂| ⊗ |ζ_n>< ζ_n|)Bv₂⊗ζ

=u₁⊗ξ, A(|v₁>< u₂| ⊗1_H)Bv₂⊗ζ. Thus it follows that

A(u₁, v₁)B(u₂, v₂) = [A(|v₁>< u₂| ⊗1_H)B](u₁, v₂).

(iv) By part (iii)

N j=1

A(e_j, u)ξ ²

= N j=1

ξ, A^∗(u, e_j)A(e_j, u)ξ

=ξ,[A^∗(P_N⊗1_H)A](u, u)ξ, whereP_N is the finite rank projection_N

j=1|e_j >< e_j|onh.Since{[A^∗(P_N⊗ 1_H)A](u, u)}is an increasing sequence of positive operators and 0≤P_N ⊗1_H converges strongly to 1_h⊗HasN tends to∞,[A^∗(P_N ⊗1_H)A](u, u) converges strongly to [A^∗A](u, u) asN tends to∞.Thus

N→∞lim N j=1

A(e_j, u)ξ ²=ξ,[A^∗A](u, u)ξ

and N j=1

A(e_j, u)ξ ²≤ A u⊗ξ ²≤ A ² u ² ξ ²,∀N ≥1.

Now let us consider the following, forξ, ζ∈ H

|ξ, N j=1

A(u, e_j)B(e_j, v)ζ|²=|^N

j=1

A^∗(e_j, u)ξ, B(e_j, v)ζ|²

≤ N j=1

A^∗(e_j, u)ξ ² N j=1

B(e_j, v)ζ ²

≤ A ² u ² ξ ² B ² v ² ζ ². So

|ξ, N j=1

A(u, e_j)B(e_j, v)ζ| ≤ A B u v ξ ζ and strong convergence of

j≥1A(u, e_j)B(e_j, v) follows.

(v) We have

ξ, A(u, v)^∗A(u, v)ξ=

n≥1

ξ, A^∗(v, u)ζ_nζ_n, A(u, v)ξ

=

n≥1

v⊗ξ, A^∗u⊗ζ_nu⊗ζ_n, Av⊗ξ

=v⊗ξ, A^∗{|u >< u| ⊗

n≥1

|ζ_n>< ζ_n|}Av⊗ξ. Since

n≥1|ζ_n>< ζ_n| converges strongly to the identity operator ξ, A(u, v)^∗A(u, v)ξ ≤ u ²v⊗ξ, A^∗Av⊗ξ and this proves the result.

(vi) We have

A(u, v)ξ₁, B(p, w)ξ₂

=

n≥1

A(u, v)ξ₁, ζ_nζ_n, B(p, w)ξ₂

=

n≥1

Av⊗ξ₁, u⊗ζ_np⊗ζ_n, Bw⊗ξ₂

=

n≥1

B^∗p⊗ζ_n, w⊗ξ₂v⊗ξ₁, A^∗u⊗ζ_n

=

n≥1

p⊗ζ_n, B(|w >< v| ⊗ |ξ₂>< ξ₁|)A^∗u⊗ζ_n.

This proves the first part. The other part follows from

n≥1

p⊗ζ_n, B(|w >< v| ⊗ |ξ₂>< ξ₁|)A^∗u⊗ζ_n

=T r_h⊗H[(|u >< p| ⊗1_H)B(|w >< v| ⊗ |ξ₂>< ξ₁|)A^∗]

=T r_h⊗H[(|w >< v| ⊗ |ξ₂>< ξ₁|)A^∗(|u >< p| ⊗1_H)B]

=v⊗ξ₁, [A^∗(|u >< p| ⊗1_H)Bw⊗ξ₂

Let us briefly recall the fundamental integrator processes of quantum stochastic calculus and the flow equation, introduced by Hudson and Parthasarathy [6]. For a Hilbert space k let us consider the symmetric Fock space Γ = Γ(L²(R+,k)).The exponential vector in the Fock space, associated with a vectorf ∈L²(R+,k) is given by

e(f) =

n≥0

√1 n!f⁽ⁿ⁾, where f⁽ⁿ⁾ = f⊗f ⊗ · · · ⊗f

n−copies

for n > 0 and by convention f⁽⁰⁾ = 1. The exponential vectore(0) is called the vacuum vector.

Let us consider the Hudson-Parthasarathy (HP) flow equation on h⊗ Γ(L²(R₊,k)):

(2.2) V_s,t= 1_h⊗Γ+

μ,ν≥0 t s

V_s,τL^μ_νΛ^ν_μ(dτ).

Here the coefficientsL^μ_ν :μ, ν ≥0 are operators inhand Λ^ν_μare fundamental processes with respect to a fixed orthonormal basis{E_j :j≥1} ofk:

(2.3) Λ^μ_ν(t) =

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

t1_h⊗Γ for (μ, ν) = (0,0) a(1_[0,t]⊗E_j) for (μ, ν) = (j,0) a^†(1_[0,t]⊗E_k) for (μ, ν) = (0, k) Λ(1_[0,t]⊗ |E_k>< E_j|) for (μ, ν) = (j, k).

These operators act non-trivially on Γ_[0,t] and as identity on Γ_(t,∞),where for any interval I ⊆ [0,∞),Γ_I = Γ_sym(L²(I)). For any 0 ≤ s ≤ t < ∞,Γ = Γ_[0,s]⊗Γ_(s,t]⊗Γ_(t,∞).

Theorem 2.2([7, 14, 16, 3]). LetH ∈ B(h)be self-adjoint,{L_k, W_k^j : j, k≥1}be a family of bounded linear operators inhsuch thatW=

j,k≥1W_k^j⊗

|E_j>< E_k|is an isometry (respectively co-isometry)operator inh⊗kand for some constantc≥0,

k≥1

L_ku ²≤c u ², ∀u∈h.

Let the coefficientsL^μ_ν be as follows,

(2.4) L^μ_ν =

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

iH−¹₂

k≥1L^∗_kL_k for(μ, ν) = (0,0)

L_j for(μ, ν) = (j,0)

−

j≥1L^∗_jW_k^j for(μ, ν) = (0, k) W_k^j−δ^j_k for(μ, ν) = (j, k).

Then there exists a unique isometry (respectively co-isometry)operator valued processV_s,t satisfying (2.2).

§3. Hilbert Tensor Algebra

For a product vector u = u₁ ⊗u₂ ⊗ · · · ⊗u_n ∈ h^⊗n we shall denote the product vector u_n ⊗u_n−1 ⊗ · · · ⊗u₁ by ←u−. For the null vector in h^⊗n we shall write 0. If {f_j}^∞_j=1 is an ONB for h, then we have a product ONB {fj=f_j1⊗ · · · ⊗f_jn:j= (j₁, j₂,· · ·, j_n), j_k≥1} for the Hilbert spaceh^⊗n.

Consider Z2 ={0,1},the finite field with addition modulo 2. Forn≥1, let Zⁿ₂ denote the n-fold direct sum of Z2. For = (₁, ₂,· · ·, _n) and = (₁, ₂,· · · , _m) we define

⊕= (₁,· · ·, _n, ₁,· · ·, _m)∈Z^n+m₂ and ^∗= (1+_n,1+_n−1,· · ·,1+₁)∈Zⁿ₂. Let A∈ B(h⊗ H), ∈Z₂ ={0,1}.We define operatorsA⁽⁾∈ B(h⊗ H) by A⁽⁾ := A if = 0 and A⁽⁾ := A^∗ if = 1. For 1 ≤ k ≤ n, we define a unitary exchange mapP_k,n:h^⊗n⊗ H →h^⊗n⊗ Hby putting

P_k,n(u⊗ξ) :=u₁⊗ · · · ⊗u_k−1⊗u_k+1· · · ⊗u_n⊗(u_k⊗ξ)

on product vectors. Let= (₁, ₂,· · · , _n)∈Zⁿ₂. Consider the ampliation of the operatorA^(k)in B(h^⊗n⊗ H) given by

A^(n,k):=P_k,n^∗ (1_h⊗(n−1)⊗A^(k))P_k,n.

By definition of unitary operatorsP_k,nfor product vectorsu,v∈h^⊗n, ξ, A^(n,k)(u,v)ξ=ξ, A^(k)(u_k, v_k)ξⁿ

j=k

u_j, v_j,∀ξ, ξ∈H.

Now we define the operator A⁽⁾ := _n

k=1 A^(n,k) := A^(1,1)· · ·A^(n,n) in B(h^⊗n⊗ H). Please note that as here, through out this article, the product symbol_n

k=1 stands for product from left to right askincreases. We have the following preliminary observation.

Lemma 3.1. (i)For product vectors u,v∈h^⊗n m

i=1

A^(n,i)(u,v) = m i=1

Aⁱ(u_i, v_i) n i=m+1

u_i, v_i ∈ B(H).

(ii) Forξ, ζ∈ H m i=1

A⁽ⁱ⁾(ξ, ζ) =A^((m))(ξ, ζ)⊗1_h⊗n−m ∈B(h^⊗n).

(iii)IfAis an isometry (respectively unitary)thenA^(n,k)andA⁽⁾are isome- tries (respectively unitaries).

The proof is obvious and is omitted.

We note that part (i) of this Lemma in particular gives

(3.1) A⁽⁾(u,v) =

n i=1

A⁽ⁱ⁾(u_i, v_i)

Let M₀ :={(u,v, ) : u = ⊗ⁿ_i=1u_i, v = ⊗ⁿ_i=1v_i ∈ h^⊗n, = (₁, ₂,· · ·, _n) ∈ Zⁿ₂, n ≥ 1}. In M₀, we introduce an equivalence relation ‘ ∼’ : (u,v, ) ∼ (p,w, ) if= and |u><v| =|p>< w| ∈ B(h^⊗n). Expanding the vectors in term of the ONB{ej =e_j1⊗ · · · ⊗e_jn : j= (j₁, j₂,· · · , j_n), j_k ≥1}, from

|u>< v| =|p>< w| we getujvk = pjwk for each multi-indices j,k. Thus in particular when (u, v,0)∼(p, w,0),for any ξ₁, ξ₂∈ Hwe have

ξ₁, A(u, v)ξ₂

=

j,k≥1

u_jv_ke_j⊗ξ₁, Ae_k⊗ξ₂

=

j,k≥1

p_jw_ke_j⊗ξ₁, Ae_k⊗ξ₂

=ξ₁, A(p, w)ξ₂.

In factA(u, v) =A(p, w) iff (u, v,0)∼(p, w,0) and more generallyA⁽⁾(u,v) = A⁽⁾(p,w) iff (u,v, ) ∼ (p,w, ). It is easy to see that (0,v, ) ∼ (u,0, ) ∼ (0,0, ) and we call this class the 0 of the quotient setM₀/∼. We first intro- duce the following operations onM₀:

Vector multiplication: (u,v, ).(p,w, ) = (u⊗p,v⊗w, ⊕) and Involution: (u,v, )^∗= (←v−,←u−, ^∗).

Since (u⊗p)

←−−−−=p_m⊗· · ·⊗p₁⊗u_n⊗· · ·⊗u₁= (p

←−⊗←u−) and (⊕)^∗= ()^∗⊕^∗ [(u,v, ).(p,w, )]^∗= (u⊗p,v⊗w, ⊕)^∗

= (v⊗w

←−−−,u⊗p

←−−−,(⊕)^∗)

= (w←− ⊗←v−, p

←−⊗←u−,()^∗⊕^∗)

= (p,w, )^∗.(u,v, )^∗.

It is clear that = =⇒ ^∗ = ()^∗ and |u >< v| = |p >< w| implies

|←v−><←u−|=|←w−>< p

←−|.Thus (u,v, )∼(p,w, ) implies (u,v, )^∗∼(p,w, )^∗. Moreover, (u,v, )∼(u,v, ) and (p,w, α)∼(p,w, α) implies⊕α=⊕α and |u⊗p >< v⊗w| = |u >< v| ⊗ |p >< w| = |u >< v| ⊗ |p >< w| =

|u⊗p><v⊗w|.Thus involution and vector multiplication respect∼. LetM be the complex vector space spanned byM₀/∼.The elements of M are formal finite linear combinations of elements ofM₀/∼.With the above multiplication and involutionM is a∗-algebra.

§4. Unitary Processes with Stationary and Independent Increment Let {U_s,t : 0≤s≤t <∞}be a family of unitary operators in B(h⊗ H) and Ω be a fixed unit vector inH. We shall also set U_t:=U_0,t for simplicity.

As we discussed in the previous section, let us consider the family of operators {U_s,t⁽⁾}inB(h⊗ H) for∈Z2given byU_s,t⁽⁾=U_s,tif= 0, U_s,t⁽⁾=U_s,t^∗ if= 1.

Furthermore forn ≥ 1, ∈ Zⁿ₂ fixed, 1 ≤ k ≤ n, we consider the families of operators{U_s,t^(k)}and{U_s,t⁽⁾}inB(h^⊗n⊗ H). By Lemma 3.1 we observe that

U_s,t⁽⁾(u,v) = n i=1

U_s,t⁽ⁱ⁾(u_i, v_i).

For= (0,0,· · ·,0)∈Zⁿ₂ and 1≤k≤n,we shall write U_s,t^(n,k)for the unitary operator U_s,t^(n,k) and U_s,t⁽ⁿ⁾ for the unitary U_s,t⁽⁾ on h^⊗n⊗ H. For n ≥ 1,s = (s₁, s₂,· · · , s_n),t= (t₁, t₂,· · · , t_n) : 0≤s₁ ≤t₁ ≤s₂ ≤. . .≤s_n ≤t_n <∞, _k = (α^(k)₁ , α^(k)₂ ,· · · , α^(k)_mk) ∈ Z^m₂^k : 1 ≤ k ≤ n, m = m₁+m₂+· · ·+m_n =₁⊕₂⊕ · · · ⊕_n ∈Z^m₂,we define Us⁽⁾,t ∈ B(h^⊗m⊗ H) by setting

(4.1) Us⁽⁾_,t :=

n k=1

U_s⁽_k^k_,t⁾_k.

Here U_s^(k)

k,tk is looked upon as an operator in B(h^⊗m⊗ H) by ampliation and appropriate tensor flip. So foru=⊗ⁿ_k=1u_k,v=⊗ⁿ_k=1v_k ∈h^⊗mwe have

Us⁽⁾,t(u,v) = n k=1

U_s⁽_k^k_,t⁾_k(u_k,v_k).

When there can be no confusion, for = (0,0,· · ·,0) we writeUs,t for Us⁽⁾_,t. For a, b ≥ 0,s = (s₁, s₂,· · ·, s_n),t = (t₁, t₂,· · · , t_n) we write a ≤ s,t ≤ b if a≤s₁≤t₁≤s₂≤. . .≤s_n≤t_n≤b.

Let us assume the following properties on the unitary familyU_s,tto estab- lish unitary equivalence ofU_s,twith an HP flow.

Assumption A

A1 (Evolution)For any 0≤r≤s≤t <∞, U_r,sU_s,t=U_r,t.

A2 (Independence of increments)For any 0≤s_i ≤t_i <∞ : i= 1,2 such that [s₁, t₁)∩[s₂, t₂) =∅

(a) For every u_i, v_i ∈ h, U_s1,t1(u₁, v₁) commutes with U_s2,t2(u₂, v₂) and U_s^∗_2,t2(u₂, v₂).

(b) Fors₁≤a,b≤t₁, s₂≤q,r≤t₂ andu,v∈h^⊗n, p,w∈h^⊗m, ∈Zⁿ₂, ∈Z^m₂

Ω, U⁽⁾

a,b(u,v)Uq⁽,r⁾(p,w)Ω=Ω, U⁽⁾

a,b(u,v)ΩΩ, Uq⁽,r⁾(p,w)Ω. A3 (Stationarity)For any 0≤s≤t <∞andu,v∈h^⊗n, ∈Zⁿ₂

Ω, U_s,t⁽⁾(u,v)Ω=Ω, U_t−s⁽⁾(u,v)Ω. Assumption B (Uniform continuity)

t→0lim sup{|Ω,(U_t−1)(u, v)Ω|: u , v = 1}= 0.

Assumption C (Gaussian Condition) For any u_i, v_i ∈ h, _i ∈ Z2 : i = 1,2,3

(4.2) lim

t→0

1

tΩ, (U_t⁽¹⁾−1)(u₁, v₁)(U_t⁽²⁾−1)(u₂, v₂)(U_t⁽³⁾−1)(u₃, v₃) Ω= 0.

Assumption D (Minimality)

The setS ={Us_,t(u,v)Ω :=U_s1,t1(u₁, v₁)· · ·U_sn,tn(u_n, v_n)Ω :s = (s₁, s₂,

· · ·, s_n),t= (t₁, t₂,· · · , t_n) : 0≤s₁≤t₁≤s₂≤. . .≤s_n ≤t_n<∞, n≥ 1,u=⊗ⁿ_i=1u_i,v=⊗ⁿ_i=1v_i∈h^⊗n}is total inH.

Remark 4.1. (a) TheAssumptions A, B andC hold in many situa- tions, for example for unitary solutions of the Hudson-Parthasarathy flow (2.2) with bounded operator coefficients and having no Poisson terms. We will see (Lemma 6.6) thatAssumption C means that expressions of the form (4.2) vanish for arbitrary lengthn ≥3 not only for length 3. This corresponds to the fact that the increments of ordern≥3 of a Gaussian distribution vanish.

Moreover, this property characterizes Gaussian distributions.

In the case ofdim(h)<∞Assumption Cis equivalent to the condition that the generator of the quantum L´evy process associated with {U_s,t} van- ishes on products of length 3 of elements of the kernel of the counit. This again is equivalent to the condition that the preservation term in the corresponding quantum stochastic differential equation does not appear, Ref. Chapter 5 of [19].

Remark 4.2. TheAssumption D is not really a restriction, since one can as well work with replacingHby the span closure of S. Taking 0≤s₁ ≤ t₁ ≤ s₂ ≤ . . . ≤ s_n ≤ t_n < ∞ in the definition of S ⊆ H is enough for totality of the setS because : for 0 ≤r ≤s≤t≤ ∞, we have U_r,t(p, w)) =

jU_r,s(p, e_j)U_s,t(e_j, w). So if there are overlapping intervals [s_k, t_k)∩[s_k+1, t_k+1)=∅then the vectorξ=Us,t(u,v)Ω inHcan be obtained as a vector in the closure of the linear span ofS.

For anyn≥1 we have the following useful observations.

Lemma 4.3. (i)For any 0≤r≤s≤t <∞, (4.3) U_r,t^(n,k)=U_r,s^(n,k)U_s,t^(n,k).

(ii)For any1≤k₁, k₂,· · · , k_m≤n :k_i=k_j fori=j and0≤s_i≤t_i<∞ : i= 1,2,· · ·, n

(4.4)

m i=1

U_s^(n,ki)

i,ti (u,v) = m i=1

U_s^(n,ki)

i,ti (u_ki, v_ki)

j=ki

u_j, v_j

for every u=⊗ⁿ_i=1u_i,v=⊗ⁿ_i=1v_i∈h^⊗n and∈Zⁿ₂. (iii)

(4.5) U_r,t⁽ⁿ⁾=U_r,s⁽ⁿ⁾U_s,t⁽ⁿ⁾.

Proof. (i) It follows from the definition andAssumptions A1, A2.

(ii) Foru=⊗ⁿ_i=1u_i,v=⊗ⁿ_i=1v_i∈h^⊗nand∈Zⁿ₂ we can see by induction that ξ,Y^m

i=1

U_s^(n,_i,ti^ki)(u,v)ξ=ξ,Y^m

i=1

U_s^(n,_i,ti^ki)(uki, vki) Y

j=k_i

ξYⁿ

j=k

uj, vj,∀ξ, ξ∈H.

Thus (4.4) follows.

(iii) SinceU_r,t⁽ⁿ⁾is a product ofU_r,t^(n,k) :k= 1,2, . . . , nand we have U_r,t^(n,k)=U_r,s^(n,k)U_s,t^(n,k),

it is enough to prove that the unitary operatorsU_r,s^(n,k) andU_s,t^(n,l)commute for k=l.To see this let us consider the following. By part (ii) and the fact that U_r,s(u_k, v_k) andU_s,t(u_l, v_l) commute byAssumption A2, we get

U_r,s^(n,k)U_s,t^(n,l)(u,v) =U_r,s(u_k, v_k)U_s,t(u_l, v_l)

i=k,l

u_i, v_i

=U_s,t(u_l, v_l)U_r,s(u_k, v_k)

i=k,l

u_i, v_i=U_s,t^(n,l)U_r,s^(n,k)(u,v).

As all the operatorsU appear here are bounded this implies U_r,s^(n,k)U_s,t^(n,l)=U_s,t^(n,l)U_r,s^(n,k).

§5. Filtration

For any 0≤q≤t <∞,letH_[q,t] =SpanS_[q,t],where S_[q,t] ⊆ His given by {ξ_[q,t] = Ur⁽ⁿ⁾_,s(u,v)Ω = U_r1,s1(u₁, v₁)· · ·U_rn,sn(u_n, v_n)Ω ∈ S : q ≤ r,s <

t, n≥1,u,v∈ h^⊗n}. We shall denote the Hilbert spaces H_[0,t] and H_[t,∞) by H_t] andH_[trespectively.

Lemma 5.1. For 0 ≤t ≤ T ≤ ∞, there exist a unitary isomorphism Ξ_t:H_t]⊗ H_(t,T]→ H_T] such that

(5.1) U_t(u, v) = Ξ^∗_tU_t(u, v)⊗1_H(t,T]Ξ_t. Proof. Let us define a map Ξ_t:H_t]⊗ H_[t,T]→ H_T] by

Ξ_t(ξ_[0,t]⊗ζ_[t,T]) =Ur⁽ⁿ⁾,s(u,v)Ur⁽ⁿ⁾_,s(p,w)Ω

forξ_[0,t] =Ur⁽ⁿ⁾,s(u,v)Ω∈ S_t] andζ_[t,T] =Ur⁽ⁿ⁾_,s(p,w)Ω∈ S_[t,T], then extending linearly.

Now let us consider the following. ByAssumption A, forξ_[0,t] andζ_[t,T] as above andη_[0,t] =Ua⁽ⁿ⁾_,b(x,y)Ω ∈ S_t] and γ_[t,T] =U⁽ⁿ⁾

a,b(g,h)Ω ∈ S_[t,T], we have

Ξ_t(ξ_[0,t]⊗ζ_[t,T]),Ξ_t(η_[0,t]⊗γ_[t,T])

=Ur⁽ⁿ⁾_,s(u,v)Ur⁽ⁿ⁾_,s(p,w)Ω, Ua⁽ⁿ⁾_,b(x,y)U⁽ⁿ⁾

a,b(g,h)Ω

=Ω,

Ur⁽ⁿ⁾_,s(u,v)Ur⁽ⁿ⁾,s(p⁽ⁿ⁾,w⁽ⁿ⁾) _∗

Ua⁽ⁿ⁾_,b(x,y)U⁽ⁿ⁾

a,b(g,h)Ω

=Ω,

Ur⁽ⁿ⁾_,s(u,v) _∗

Ua⁽ⁿ⁾_,b(x,y)Ω Ω,

Ur⁽ⁿ⁾,s(p,w) _∗

U⁽ⁿ⁾

a_,b(g,h)Ω

=ξ_[0,t], η_[0,t]ζ_[t,T], γ_[t,T].

Thus we getΞ_t(ξ_[0,t]⊗ζ_[t,T]),Ξ_t(η_[0,t]⊗γ_[t,T])=ξ_[0,t]⊗ζ_[t,T], η_[0,t]⊗γ_[t,T]. Since by definition the range of Ξ_t is dense inH_T],this proves Ξ_tis a unitary operator.

Again by similar arguments to those above, for any u, v∈h, we have Ξ_tξ_[0,t]⊗ζ_[t,T], U_t(u, v) Ξ_t η_[0,t]⊗γ_[t,T]

=Ur⁽ⁿ⁾_,s(u,v)Ω, U_t(u, v)Ua⁽ⁿ⁾_,b(x,y)Ω Ur⁽ⁿ⁾,s(p,w)Ω, U⁽ⁿ⁾

a_,b(g,h)Ω

=ξ_[0,t] , U_t(u, v)η_[0,t] ζ_[t,T], γ_[t,T]. This proves (5.1).

§6. Expectation Semigroups

Let us look at the various semigroups associated with the unitary evolution {U_s,t}.For any fixedn≥1,we define a family of operators {T_t⁽ⁿ⁾} onh^⊗n by setting

φ, T_t⁽ⁿ⁾ψ:=Ω, U_t⁽ⁿ⁾(φ, ψ) Ω, ∀φ, ψ∈h^⊗n. Then in particular for product vectorsu=⊗ⁿ_i=1u_i, v=⊗ⁿ_i=1v_i ∈h^⊗n

u, T_t⁽ⁿ⁾ v=Ω, U_t⁽ⁿ⁾(u,v) Ω=Ω, U_t(u₁, v₁)U_t(u₂, v₂)· · ·U_t(u_n, v_n) Ω. Forn= 1,we shall writeT_tfor the familyT_t⁽¹⁾.

Lemma 6.1. The above family of operators {T_t⁽ⁿ⁾} is a semigroup of contractions onh^⊗n.

Proof. SinceU_t⁽ⁿ⁾is in particular contractive, for anyφ, ψ∈h^⊗n

|φ, T_t⁽ⁿ⁾ψ|=|φΩ, U_t⁽ⁿ⁾ψΩ| ≤ φ ψ and contractivity ofT_t⁽ⁿ⁾follows.

In order to prove that this family of contractionsT_t⁽ⁿ⁾is a semigroup it is enough to show that for any 0≤s ≤t and product vectors u=⊗ⁿ_i=1u_i, v=

⊗ⁿ_i=1v_i∈h^⊗n,

u, T_t⁽ⁿ⁾v=u, T_s⁽ⁿ⁾T_t−s⁽ⁿ⁾v.

Consider the product orthonormal basis {ej = e_j1 ⊗e_j2 ⊗ · · · ⊗e_jn : j = (j₁, j₂,· · · , j_n) :j₁, j₂,· · ·, j_n ≥1}ofh^⊗n. By part (iii) of Lemma 2.1 and the evolution property (4.5) ofU_t⁽ⁿ⁾,

u, T_t⁽ⁿ⁾v=Ω, U_t⁽ⁿ⁾(u,v) Ω

=

j

Ω, U_s⁽ⁿ⁾(u, ej)U_s,t⁽ⁿ⁾(ej,v)Ω

=

j

Ω, U_s⁽ⁿ⁾(u, ej) ΩΩ, U_t−s⁽ⁿ⁾(ej,v)Ω

=

j

u, T_s⁽ⁿ⁾ejej, T_t−s⁽ⁿ⁾v=u, T_s⁽ⁿ⁾T_t−s⁽ⁿ⁾v.

The following Lemma will be needed in the sequel Lemma 6.2. (i)For1≤k≤n,

(6.1) Ω, U_t^(n,k)(p,w)Ω=p, T_t^(n,k)w, ∀p,w∈h^⊗n whereT_t^(n,k)= 1_h(⊗k−1)⊗T_t⊗1_h(⊗n−k).

(ii) For any 1≤m≤n, p,w∈h^⊗n, Ω,(

m k=1

U_t^(n,k))(p,w)Ω=p, T_t^(m)⊗1_h(⊗n−m) w.

(iii) For anyφ∈h^⊗n,

(U_t^(n,k)−1)φ⊗Ω ²

=(1−T_t^(n,k))φ, φ+φ,(1−T_t^(n,k))φ

≤2 1−T_t φ ².