**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*_{t}*L*^{μ}* _{ν}*Λ

^{ν}*(dt), V*

_{μ}_{0}= 1

_{h⊗Γ}*,*

(where the coeﬃcients *L*^{μ}* _{ν}* :

*μ, ν*

*≥*0 are operators in the initial Hilbert space

**h**and Λ

^{ν}*are fundamental processes in the symmetric Fock space Γ = Γ*

_{μ}*(L*

_{sym}^{2}(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* ^{th}*Mile, 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 on**h***⊗ H,***h**ﬁnite 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 on**h***⊗ H,***h**not 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
spaces*H,K B*(*H,K*) and*B*_{1}(*H*) denote the Banach space of bounded linear
operators from*H*to*K*and trace class operators on*H*respectively. For a linear
(not necessarily bounded) map*T* we write its domain as*D*(T).We denote the
trace on*B*1(*H*) by *T r** _{H}* or simply

*T r.*The von Neumann algebra of bounded linear operators on

*H*is denoted by

*B(H*).The Banach space

*B*1(

*H,K*)

*≡ {ρ∈*

*B*(

*H,K*) :

*|ρ|*:=

*√*

*ρ*^{∗}*ρ∈ B*_{1}(*H*)*}*with norm (Ref. Page no. 47 in [2])
*ρ* _{1}=* |ρ| *_{B}_{1}_{(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*), *B*_{1}(*H,K*) *ρ* *→*
*T r** _{H}*(xρ) deﬁnes an element of the dual Banach space

*B*

_{1}(

*H,K*)

^{∗}*.*For a linear map

*T*on the Banach space

*B*

_{1}(

*H,K*) the adjoint

*T*

*on the dual*

^{∗}*B*(

*K,H*) is given by

*T r*

*(T*

_{H}*(x)ρ) :=*

^{∗}*T r*

*(xT(ρ)),*

_{H}*∀x∈ B*(

*K,H*), ρ

*∈ B*1(

*H,K*).

For any *ξ∈ H ⊗ K, h∈ H*the map

*K k→ ξ, h⊗k*

deﬁnes a bounded linear functional on *K* and thus by Riesz’s theorem there
exists a unique vector*ξ, h*in*K* such that

(2.1) *ξ, h, k*=*ξ, h⊗k,∀k∈ K.*

In other words*ξ, h*=*F*_{h}^{∗}*ξ*where*F*_{h}*∈ B*(*K,H ⊗K*) is given by*F*_{h}*k*=*h⊗k.*

Let**h**and*H*be 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∈*

**h**we deﬁne a linear operator

*A(u, v)∈ B*(

*H*) by

*ξ*_{1}*, A(u, v)ξ*_{2}=*u⊗ξ*_{1}*, A v⊗ξ*_{2}*,* *∀ξ*_{1}*, ξ*_{2}*∈ 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
*in***h**

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

*A*

*(v, u).*

^{∗}**(ii) h***×***h***→A(·,·*)*is*1*−*1, *i.e. ifA(u, v) =B(u, v),* *∀u, v∈***h***thenA*=*B.*

**(iii)** *A(u*_{1}*, v*_{1})B(u_{2}*, v*_{2}) = [A(*|v*_{1}*>< u*_{2}*| ⊗*1* _{H}*)B](u

_{1}

*, v*

_{2})

**(iv)**

*AB(u, v) =*

*j≥1**A(u, e** _{j}*)B(e

_{j}*, v) (strongly)*

**(v)**0

*≤A(u, v)*

^{∗}*A(u, v)≤ u*

^{2}

*A*

^{∗}*A(v, v)*

**(vi)** *A(u, v)ξ*_{1}*, B(p, w)ξ*_{2}=

*n≥1**p⊗ζ*_{n}*,*[B(*|w >< v|⊗|ξ*_{2}*>< ξ*_{1}*|*)A^{∗}*u⊗ζ*_{n}

=*v⊗ξ*_{1}*,* [A* ^{∗}*(

*|u >< p| ⊗*1

*)Bw*

_{H}*⊗ξ*

_{2}

*.*

*Proof.*We are omitting the proof of (i), (ii).

(iii) For any*ξ, ζ∈ H*we have

*ξ, A(u*_{1}*, v*_{1})B(u_{2}*, v*_{2})ζ=*u*_{1}*⊗ξ, Av*_{1}*⊗B(u*_{2}*, v*_{2})ζ

=*A*^{∗}*u*_{1}*⊗ξ, v*_{1}*⊗B(u*_{2}*, v*_{2})ζ

=

*n≥1*

*A*^{∗}*u*_{1}*⊗ξ, v*_{1}*⊗ζ*_{n}*ζ*_{n}*, B(u*_{2}*, v*_{2})ζ

=

*n≥1*

*A*^{∗}*u*_{1}*⊗ξ, v*_{1}*⊗ζ*_{n}*u*_{2}*⊗ζ*_{n}*, Bv*_{2}*⊗ζ*

=

*n≥1*

*A*^{∗}*u*_{1}*⊗ξ,*(*|v*_{1}*>< u*_{2}*| ⊗ |ζ*_{n}*>< ζ*_{n}*|*)Bv_{2}*⊗ζ*

=*u*_{1}*⊗ξ, A(|v*_{1}*>< u*_{2}*| ⊗*1* _{H}*)Bv

_{2}

*⊗ζ.*Thus it follows that

*A(u*_{1}*, v*_{1})B(u_{2}*, v*_{2}) = [A(*|v*_{1}*>< u*_{2}*| ⊗*1* _{H}*)B](u

_{1}

*, v*

_{2}).

(iv) By part (iii)

*N*
*j=1*

*A(e*_{j}*, u)ξ* ^{2}

=
*N*
*j=1*

*ξ, A** ^{∗}*(u, e

*)A(e*

_{j}

_{j}*, u)ξ*

=*ξ,*[A* ^{∗}*(P

_{N}*⊗*1

*)A](u, u)ξ*

_{H}*,*where

*P*

*is the ﬁnite rank projection*

_{N}

_{N}*j=1**|e*_{j}*>< e*_{j}*|*on**h.**Since*{*[A* ^{∗}*(P

_{N}*⊗*1

*)A](u, u)*

_{H}*}*is an increasing sequence of positive operators and 0

*≤P*

_{N}*⊗*1

*converges strongly to 1*

_{H}**as**

_{h⊗H}*N*tends to

*∞,*[A

*(P*

^{∗}

_{N}*⊗*1

*)A](u, u) converges strongly to [A*

_{H}

^{∗}*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, for*ξ, 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≥1**A(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

_{n}*ζ*

_{n}*, A(u, v)ξ*

=

*n≥1*

*v⊗ξ, A*^{∗}*u⊗ζ*_{n}*u⊗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* ^{2}*v⊗ξ, A*^{∗}*Av⊗ξ*
and this proves the result.

(vi) We have

*A(u, v)ξ*_{1}*, B(p, w)ξ*_{2}

=

*n≥1*

*A(u, v)ξ*_{1}*, ζ*_{n}*ζ*_{n}*, B(p, w)ξ*_{2}

=

*n≥1*

*Av⊗ξ*_{1}*, u⊗ζ*_{n}*p⊗ζ*_{n}*, Bw⊗ξ*_{2}

=

*n≥1*

*B*^{∗}*p⊗ζ*_{n}*, w⊗ξ*_{2}*v⊗ξ*_{1}*, A*^{∗}*u⊗ζ*_{n}

=

*n≥1*

*p⊗ζ*_{n}*, B(|w >< v| ⊗ |ξ*_{2}*>< ξ*_{1}*|*)A^{∗}*u⊗ζ*_{n}*.*

This proves the ﬁrst part. The other part follows from

*n≥1*

*p⊗ζ*_{n}*, B(|w >< v| ⊗ |ξ*_{2}*>< ξ*_{1}*|*)A^{∗}*u⊗ζ*_{n}

=*T r*** _{h⊗H}**[(

*|u >< p| ⊗*1

*)B(*

_{H}*|w >< v| ⊗ |ξ*

_{2}

*>< ξ*

_{1}

*|*)A

*]*

^{∗}=*T r*** _{h⊗H}**[(

*|w >< v| ⊗ |ξ*

_{2}

*>< ξ*

_{1}

*|*)A

*(*

^{∗}*|u >< p| ⊗*1

*)B]*

_{H}=*v⊗ξ*_{1}*,* [A* ^{∗}*(

*|u >< p| ⊗*1

*)Bw*

_{H}*⊗ξ*

_{2}

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(L^{2}(R+*,***k)).**The exponential vector in the Fock space, associated
with a vector*f* *∈L*^{2}(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 vector**e(0) is called the vacuum vector.**

Let us consider the Hudson-Parthasarathy (HP) ﬂow equation on **h***⊗*
Γ(L^{2}(R_{+}*,***k)):**

(2.2) *V** _{s,t}*= 1

**+**

_{h⊗Γ}*μ,ν≥0*
*t*
*s*

*V*_{s,τ}*L*^{μ}* _{ν}*Λ

^{ν}*(dτ).*

_{μ}Here the coeﬃcients*L*^{μ}* _{ν}* :

*μ, ν*

*≥*0 are operators in

**h**and Λ

^{ν}*are fundamental processes with respect to a ﬁxed orthonormal basis*

_{μ}*{E*

*:*

_{j}*j≥*1

*}*of

**k**:

(2.3) Λ^{μ}* _{ν}*(t) =

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

*t*1** _{h⊗Γ}** for (μ, ν) = (0,0)

*a(1*

_{[0,t]}

*⊗E*

*) for (μ, ν) = (j,0)*

_{j}*a*

*(1*

^{†}_{[0,t]}

*⊗E*

*) for (μ, ν) = (0, k) Λ(1*

_{k}_{[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

*(L*

_{sym}^{2}(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 in*

**h**

*such thatW*=

*j,k≥1**W*_{k}^{j}*⊗*

*|E*_{j}*>< E*_{k}*|is an isometry* (respectively co-isometry)*operator in***h***⊗***k***and for*
*some constantc≥*0,

*k≥1*

*L*_{k}*u* ^{2}*≤c* *u* ^{2}*,* *∀u∈***h.**

*Let the coeﬃcientsL*^{μ}_{ν}*be as follows,*

(2.4) *L*^{μ}* _{ν}* =

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

*iH−*^{1}_{2}

*k≥1**L*^{∗}_{k}*L*_{k}*for*(μ, ν) = (0,0)

*L*_{j}*for*(μ, ν) = (j,0)

*−*

*j≥1**L*^{∗}_{j}*W*_{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* = *u*_{1} *⊗u*_{2} *⊗ · · · ⊗u*_{n}*∈* **h*** ^{⊗n}* we shall denote
the product vector

*u*

_{n}*⊗u*

_{n−1}*⊗ · · · ⊗u*

_{1}by

*←u−.*For the null vector in

**h**

*we shall write 0. If*

^{⊗n}*{f*

_{j}*}*

^{∞}*is an ONB for*

_{j=1}**h,**then we have a product ONB

*{fj*=

*f*

_{j}_{1}

*⊗ · · · ⊗f*

_{j}*:*

_{n}*j*= (j

_{1}

*, j*

_{2}

*,· · ·, j*

*), j*

_{n}

_{k}*≥*1

*}*for the Hilbert space

**h**

^{⊗n}*.*

Consider Z2 =*{*0,1*},*the ﬁnite ﬁeld with addition modulo 2. For*n≥*1,
let Z^{n}_{2} denote the *n-fold direct sum of* Z2*.* For = (_{1}*, *_{2}*,· · ·, ** _{n}*) and

*= (*

^{}

^{}_{1}

*,*

^{}_{2}

*,· · ·*

*,*

^{}*) we deﬁne*

_{m}*⊕** ^{}*= (

_{1}

*,· · ·,*

_{n}*,*

^{}_{1}

*,· · ·,*

^{}*)*

_{m}*∈Z*

^{n+m}_{2}and

*= (1+*

^{∗}

_{n}*,*1+

_{n−1}*,· · ·,*1+

_{1})

*∈Z*

^{n}_{2}

*.*Let

*A∈ B*(h

*⊗ H*),

*∈*Z

_{2}=

*{*0,1

*}.*We deﬁne operators

*A*

^{()}

*∈ B*(h

*⊗ H*) by

*A*

^{()}:=

*A*if = 0 and

*A*

^{()}:=

*A*

*if = 1. For 1*

^{∗}*≤*

*k*

*≤*

*n,*we deﬁne a unitary exchange map

*P*

*:*

_{k,n}**h**

^{⊗n}*⊗ H →*

**h**

^{⊗n}*⊗ H*by putting

*P** _{k,n}*(u

*⊗ξ) :=u*

_{1}

*⊗ · · · ⊗u*

_{k−1}*⊗u*

_{k+1}*· · · ⊗u*

_{n}*⊗*(u

_{k}*⊗ξ)*

on product vectors. Let= (_{1}*, *_{2}*,· · ·* *, ** _{n}*)

*∈*Z

^{n}_{2}

*.*Consider the ampliation of the operator

*A*

^{(}

^{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 operators*P** _{k,n}*for product vectors

*u,v∈*

**h**

^{⊗n}*,*

*ξ, A*

^{(n,}

^{k}^{)}(u,

*v)ξ*

*=*

^{}*ξ, A*

^{(}

^{k}^{)}(u

_{k}*, v*

*)ξ*

_{k}

^{}

^{n}*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 as*k*increases. 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*

*A*^{}* ^{i}*(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 *M*_{0} :=*{*(u,*v, ) :* *u* = *⊗*^{n}_{i=1}*u*_{i}*,* *v* = *⊗*^{n}_{i=1}*v*_{i}*∈* **h**^{⊗n}*, *= (_{1}*, *_{2}*,· · ·, ** _{n}*)

*∈*Z

^{n}_{2}

*, n*

*≥*1

*}.*In

*M*

_{0}

*,*we introduce an equivalence relation ‘

*∼*’ : (u,

*v, )*

*∼*(p,

*w,*

*) if=*

^{}*and*

^{}*|u><v|*=

*|p><*

*w| ∈ B*(h

*). Expanding the vectors in term of the ONB*

^{⊗n}*{ej*=

*e*

_{j}_{1}

*⊗ · · · ⊗e*

_{j}*:*

_{n}*j*= (j

_{1}

*, j*

_{2}

*,· · ·*

*, j*

*), j*

_{n}

_{k}*≥*1

*}*, from

*|u><* *v|* =*|p><* *w|* we get*u*j*v*k = *p*j*w*k for each multi-indices j*,*k. Thus in
particular when (u, v,0)*∼*(p, w,0),for any *ξ*_{1}*, ξ*_{2}*∈ H*we have

*ξ*_{1}*, A(u, v)ξ*_{2}

=

*j,k≥1*

*u*_{j}*v*_{k}*e*_{j}*⊗ξ*_{1}*, Ae*_{k}*⊗ξ*_{2}

=

*j,k≥1*

*p*_{j}*w*_{k}*e*_{j}*⊗ξ*_{1}*, Ae*_{k}*⊗ξ*_{2}

=*ξ*_{1}*, A(p, w)ξ*_{2}*.*

In fact*A(u, v) =A(p, w) iﬀ (u, v,*0)*∼*(p, w,0) and more generally*A*^{()}(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 set

*M*

_{0}

*/∼.*We ﬁrst intro- duce the following operations on

*M*

_{0}:

Vector multiplication: (u,*v, ).(p,w, ** ^{}*) = (u

*⊗p,v⊗w, ⊕*

*) and Involution: (u,*

^{}*v, )*

*= (*

^{∗}*←v−,←u−,*

*).*

^{∗}Since (u*⊗p)*

*←−−−−*=*p*_{m}*⊗· · ·⊗p*_{1}*⊗u*_{n}*⊗· · ·⊗u*_{1}= (*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*∼.*
Let*M* be the complex vector space spanned by*M*_{0}*/∼.*The elements of
*M* are formal ﬁnite linear combinations of elements of*M*_{0}*/∼.*With the above
multiplication and involution*M* 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 in

*H.*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}^{()}*}*in*B*(h*⊗ H*) for*∈*Z2given by*U*_{s,t}^{()}=*U** _{s,t}*if= 0, U

_{s,t}^{()}=

*U*

_{s,t}*if= 1.*

^{∗}Furthermore for*n* *≥* 1, *∈* Z^{n}_{2} ﬁxed, 1 *≤* *k* *≤* *n,* we consider the families of
operators*{U*_{s,t}^{(}^{k}^{)}*}*and*{U*_{s,t}^{()}*}*in*B*(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)*∈*Z^{n}_{2} 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* =
(s_{1}*, s*_{2}*,· · ·* *, s** _{n}*),

*t*= (t

_{1}

*, t*

_{2}

*,· · ·*

*, t*

*) : 0*

_{n}*≤s*

_{1}

*≤t*

_{1}

*≤s*

_{2}

*≤. . .≤s*

_{n}*≤t*

_{n}*<∞,*

*= (α*

_{k}^{(k)}

_{1}

*, α*

^{(k)}

_{2}

*,· · ·*

*, α*

^{(k)}

_{m}*)*

_{k}*∈*Z

^{m}_{2}

*: 1*

^{k}*≤*

*k*

*≤*

*n, m*=

*m*

_{1}+

*m*

_{2}+

*· · ·*+

*m*

*=*

_{n}_{1}

*⊕*

_{2}

*⊕ · · · ⊕*

_{n}*∈*Z

^{m}_{2}

*,*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**,t**k* is looked upon as an operator in *B*(h^{⊗m}*⊗ H*) by ampliation and
appropriate tensor ﬂip. So for*u*=*⊗*^{n}_{k=1}*u*_{k}*,v*=*⊗*^{n}_{k=1}*v*_{k}*∈***h*** ^{⊗m}*we 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 write*Us**,**t* for *Us*^{()}_{,}*t.*
For *a, b* *≥* 0,*s* = (s_{1}*, s*_{2}*,· · ·, s** _{n}*),

*t*= (t

_{1}

*, t*

_{2}

*,· · ·*

*, t*

*) we write*

_{n}*a*

*≤*

*s,t*

*≤*

*b*if

*a≤s*

_{1}

*≤t*

_{1}

*≤s*

_{2}

*≤. . .≤s*

_{n}*≤t*

_{n}*≤b.*

Let us assume the following properties on the unitary family*U** _{s,t}*to estab-
lish unitary equivalence of

*U*

*with an HP ﬂow.*

_{s,t}**Assumption A**

**A1 (Evolution)**For any 0*≤r≤s≤t <∞, U*_{r,s}*U** _{s,t}*=

*U*

_{r,t}*.*

**A2 (Independence of increments)**For any 0*≤s*_{i}*≤t*_{i}*<∞* : *i*= 1,2
such that [s_{1}*, t*_{1})*∩*[s_{2}*, t*_{2}) =∅

(a) For every *u*_{i}*, v*_{i}*∈* **h, U**_{s}_{1}_{,t}_{1}(u_{1}*, v*_{1}) commutes with *U*_{s}_{2}_{,t}_{2}(u_{2}*, v*_{2}) and
*U*_{s}^{∗}_{2}_{,t}_{2}(u_{2}*, v*_{2}).

(b) For*s*_{1}*≤a,b≤t*_{1}*, s*_{2}*≤q,r≤t*_{2} and*u,v∈***h**^{⊗n}*,* *p,w∈***h**^{⊗m}*,*
*∈*Z^{n}_{2}*, *^{}*∈*Z^{m}_{2}

Ω, U^{()}

*a**,**b*(u,*v)Uq*^{(}*,**r*^{}^{)}(p,*w)Ω*=Ω, U^{()}

*a**,**b*(u,*v)Ω*Ω, U*q*^{(}*,**r*^{}^{)}(p,*w)Ω.*
**A3 (Stationarity)**For any 0*≤s≤t <∞*and*u,v∈***h**^{⊗n}*, ∈*Z^{n}_{2}

Ω, U_{s,t}^{()}(u,*v)Ω*=Ω, U_{t−s}^{()}(u,*v)Ω.*
**Assumption B (Uniform continuity)**

*t→0*lim 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

*t*Ω, (U_{t}^{(}^{1}^{)}*−*1)(u_{1}*, v*_{1})(U_{t}^{(}^{2}^{)}*−*1)(u_{2}*, v*_{2})(U_{t}^{(}^{3}^{)}*−*1)(u_{3}*, v*_{3}) Ω= 0.

**Assumption D (Minimality)**

The set*S* =*{Us*_{,}*t*(u,*v)Ω :=U*_{s}_{1}_{,t}_{1}(u_{1}*, v*_{1})*· · ·U*_{s}_{n}_{,t}* _{n}*(u

_{n}*, v*

*)Ω :*

_{n}*s*= (s

_{1}

*, s*

_{2}

*,*

*· · ·, s** _{n}*),

*t*= (t

_{1}

*, t*

_{2}

*,· · ·*

*, t*

*) : 0*

_{n}*≤s*

_{1}

*≤t*

_{1}

*≤s*

_{2}

*≤. . .≤s*

_{n}*≤t*

_{n}*<∞, n≥*1,

*u*=

*⊗*

^{n}

_{i=1}*u*

_{i}*,v*=

*⊗*

^{n}

_{i=1}*v*

_{i}*∈*

**h**

^{⊗n}*}*is total in

*H.*

*Remark* 4.1. (a) The**Assumptions A, B** and**C** 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) that**Assumption C** means that expressions of the form (4.2)
vanish for arbitrary length*n* *≥*3 not only for length 3. This corresponds to
the fact that the increments of order*n≥*3 of a Gaussian distribution vanish.

Moreover, this property characterizes Gaussian distributions.

In the case of*dim(h)<∞***Assumption C**is 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. The**Assumption D** is not really a restriction, since one
can as well work with replacing*H*by the span closure of *S.* Taking 0*≤s*_{1} *≤*
*t*_{1} *≤* *s*_{2} *≤* *. . .* *≤* *s*_{n}*≤* *t*_{n}*<* *∞* in the deﬁnition of *S* *⊆ H* is enough for
totality of the set*S* because : for 0 *≤r* *≤s≤t≤ ∞,* we have *U** _{r,t}*(p, w)) =

*j**U** _{r,s}*(p, e

*)U*

_{j}*(e*

_{s,t}

_{j}*, w).*So if there are overlapping intervals [s

_{k}*, t*

*)*

_{k}*∩*[s

_{k+1}*,*

*t*

*)=∅then the vector*

_{k+1}*ξ*=

*Us*

*,*

*t*(u,

*v)Ω inH*can be obtained as a vector in the closure of the linear span of

*S.*

For any*n≥*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 any*1*≤k*_{1}*, k*_{2}*,· · ·* *, k*_{m}*≤n* :*k** _{i}*=

*k*

_{j}*fori*=

*j*

*and*0

*≤s*

_{i}*≤t*

_{i}*<∞*:

*i*= 1,2,

*· · ·, n*

(4.4)

*m*
*i=1*

*U*_{s}^{(n,}^{ki}^{)}

*i**,t**i* (u,*v) =*
*m*
*i=1*

*U*_{s}^{(n,}^{ki}^{)}

*i**,t**i* (u_{k}_{i}*, v*_{k}* _{i}*)

*j=k**i*

*u*_{j}*, v*_{j}

*for every u*=*⊗*^{n}_{i=1}*u*_{i}*,v*=*⊗*^{n}_{i=1}*v*_{i}*∈***h**^{⊗n}*and∈*Z^{n}_{2}*.*
**(iii)**

(4.5) *U*_{r,t}^{(n)}=*U*_{r,s}^{(n)}*U*_{s,t}^{(n)}*.*

*Proof.* (i) It follows from the deﬁnition and**Assumptions A1, A2.**

(ii) For*u*=*⊗*^{n}_{i=1}*u*_{i}*,v*=*⊗*^{n}_{i=1}*v*_{i}*∈***h*** ^{⊗n}*and

*∈*Z

^{n}_{2}we can see by induction that

*ξ,*Y

^{m}*i=1*

*U*_{s}^{(n,}_{i}_{,t}_{i}^{ki}^{)}(u,*v)ξ** ^{}*=

*ξ,*Y

^{m}*i=1*

*U*_{s}^{(n,}_{i}_{,t}_{i}^{ki}^{)}(u*k**i**, v**k**i*) Y

*j=k*_{i}

*ξ** ^{}*Y

^{n}*j=k*

*u**j**, v**j**,∀ξ, ξ*^{}*∈***H.**

Thus (4.4) follows.

(iii) Since*U*_{r,t}^{(n)}is a product of*U*_{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 operators*U*_{r,s}^{(n,k)} and*U*_{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*

*) and*

_{k}*U*

*(u*

_{s,t}

_{l}*, v*

*) commute by*

_{l}**Assumption A2, we get**

*U*_{r,s}^{(n,k)}*U*_{s,t}^{(n,l)}(u,*v) =U** _{r,s}*(u

_{k}*, v*

*)U*

_{k}*(u*

_{s,t}

_{l}*, v*

*)*

_{l}*i=k,l*

*u*_{i}*, v*_{i}

=*U** _{s,t}*(u

_{l}*, v*

*)U*

_{l}*(u*

_{r,s}

_{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 operators*U* 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 <∞,*let*H*_{[q,t]} =*SpanS*_{[q,t]}*,*where *S*_{[q,t]} *⊆ H*is given
by *{ξ*_{[q,t]} = *Ur*^{(n)}_{,}*s*(u,*v)Ω =* *U*_{r}_{1}_{,s}_{1}(u_{1}*, v*_{1})*· · ·U*_{r}_{n}_{,s}* _{n}*(u

_{n}*, v*

*)Ω*

_{n}*∈ 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]}* and

*H*

_{[t}respectively.

**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

^{∗}

_{t}*U*

*(u, v)*

_{t}*⊗*1

_{H}_{(t,T]}Ξ

_{t}*.*

*Proof.*Let us deﬁne a map Ξ

*:*

_{t}*H*

_{t]}*⊗ H*

_{[t,T}

_{]}

*→ H*

*by*

_{T]}Ξ* _{t}*(ξ

_{[0,t]}

*⊗ζ*

_{[t,T]}) =

*Ur*

^{(n)}

*,*

*s*(u,

*v)Ur*

^{(n)}

^{}

_{,}*s*

*(p,*

^{}*w)Ω*

for*ξ*_{[0,t]} =*Ur*^{(n)}*,**s*(u,*v)Ω∈ S** _{t]}* and

*ζ*

_{[t,T}

_{]}=

*Ur*

^{(n)}

^{}

_{,}*s*

*(p,*

^{}*w)Ω∈ S*

_{[t,T]}

*,*then extending linearly.

Now let us consider the following. By**Assumption A, for***ξ*_{[0,t]} and*ζ*_{[t,T}_{]}
as above and*η*_{[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

*(η*

_{t}_{[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

*(η*

_{t}_{[0,t]}

*⊗γ*

_{[t,T]})=

*ξ*

_{[0,t]}

*⊗ζ*

_{[t,T}

_{]}

*, η*

_{[0,t]}

*⊗γ*

_{[t,T}

_{]}

*.*Since by deﬁnition the range of Ξ

*is dense in*

_{t}*H*

_{T}_{]}

*,*this proves Ξ

*is a unitary operator.*

_{t}Again by similar arguments to those above, for any *u, v∈***h,** we have
Ξ_{t}*ξ*_{[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)U

*a*

^{(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 ﬁxed*n≥*1,we deﬁne a family of operators *{T*_{t}^{(n)}*}* on**h*** ^{⊗n}* by
setting

*φ, T*_{t}^{(n)}*ψ*:=Ω, U_{t}^{(n)}(φ, ψ) Ω*,* *∀φ, ψ∈***h**^{⊗n}*.*
Then in particular for product vectors*u*=*⊗*^{n}_{i=1}*u*_{i}*,* *v*=*⊗*^{n}_{i=1}*v*_{i}*∈***h**^{⊗n}

*u, T*_{t}^{(n)} *v*=Ω, U_{t}^{(n)}(u,*v) Ω*=Ω, U* _{t}*(u

_{1}

*, v*

_{1})U

*(u*

_{t}_{2}

*, v*

_{2})

*· · ·U*

*(u*

_{t}

_{n}*, v*

*) Ω*

_{n}*.*For

*n*= 1,we shall write

*T*

*for the family*

_{t}*T*

_{t}^{(1)}

*.*

**Lemma 6.1.** *The above family of operators* *{T*_{t}^{(n)}*}* *is a semigroup of*
*contractions on***h**^{⊗n}*.*

*Proof.* Since*U*_{t}^{(n)}is in particular contractive, for any*φ, ψ∈***h**^{⊗n}

*|φ, T*_{t}^{(n)}*ψ|*=*|φ*Ω, U_{t}^{(n)}*ψ*Ω*| ≤ φ ψ*
and contractivity of*T*_{t}^{(n)}follows.

In order to prove that this family of contractions*T*_{t}^{(n)}is a semigroup it is
enough to show that for any 0*≤s* *≤t* and product vectors *u*=*⊗*^{n}_{i=1}*u*_{i}*,* *v*=

*⊗*^{n}_{i=1}*v*_{i}*∈***h**^{⊗n}*,*

*u, T*_{t}^{(n)}*v*=*u, T*_{s}^{(n)}*T*_{t−s}^{(n)}*v.*

Consider the product orthonormal basis *{ej* = *e*_{j}_{1} *⊗e*_{j}_{2} *⊗ · · · ⊗e*_{j}* _{n}* :

*j*= (j

_{1}

*, j*

_{2}

*,· · ·*

*, j*

*) :*

_{n}*j*

_{1}

*, j*

_{2}

*,· · ·, j*

_{n}*≥*1

*}*of

**h**

^{⊗n}*.*By part (iii) of Lemma 2.1 and the evolution property (4.5) of

*U*

_{t}^{(n)},

*u, T*_{t}^{(n)}*v*=Ω, U_{t}^{(n)}(u,*v) Ω*

=

*j*

Ω, U_{s}^{(n)}(u, e*j*)U_{s,t}^{(n)}(e*j,v)Ω*

=

*j*

Ω, U_{s}^{(n)}(u, e*j*) ΩΩ*, U*_{t−s}^{(n)}(e*j,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)***For*1*≤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

**(⊗n−k)**

_{h}*.*

**(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}*.*