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SOLVING QUANTUM STOCHASTIC DIFFERENTIALEQUATIONS WITH UNBOUNDED COEFFICIENTS

FRANCO FAGNOLA AND STEPHEN J WILLS

Abstract. We demonstrate a method for obtaining strong solutions to theright Hudson-Parthasarathy quantum stochastic differential equation

dUt = F αβ Ut dΛβ

α(t), U0 = 1

where U is a contraction operator process, and the matrix of coefficients [F αβ ]

consists of unbounded operators. This is achieved whenever there is a positive

self-adjoint reference operator C that behaves well with respect to the F αβ , al-

lowing us to prove that Dom C1/2 is left invariant by the operators Ut, therebygiving rigorous meaning to the formal expression above.

We give conditions under which the solution U is an isometry or coisometry

process, and apply these results to construct unital ∗-homomorphic dilationsof (quantum) Markov semigroups arising in probability and physics.

0. Introduction

A quantum Markovian cocycle is a family of ∗-homomorphisms (jt : A → C)t≥0

between two operator algebras, A ⊂ C, where C is equipped with a semigroup(σt)t≥0 of ∗-homomorphisms, and such that

js+t = s ◦ σs ◦ jt for all s, t ≥ 0, (0.1)

where s denotes an extension of js whose domain includes σs

(⋃t≥0 jt(A)

). Such

perturbations of the semigroup evolution law occur naturally in stochastic settingswhere A and C are commutative algebras of bounded measurable functions. Byno longer insisting that A and C be commutative we obtain a quantum stochasticprocess (in the sense of [AFL]) which generalises the classical notion of a flow to aform more suitable for modelling situations in quantum physics.

The most commonly studied examples of cocycles are those for which A is a ∗-subalgebra of B(h), the algebra of all bounded operators on some Hilbert space h,and C is of the form A′′ ⊗B(F), where A′′ is the von Neumann algebra generatedby A and F is the symmetric Fock space over L2(R+; k), the square integrablefunctions on R+ taking values in some Hilbert space k. Fock space here playsthe role of Wiener space in the classical theory, and is equipped with a semigroup(σt)t≥0 induced by the natural time shift on L2(R+; k). That this is an appropriatechoice for the image algebra C can be justified by a limiting procedure motivated byphysical arguments (see, for example, [AAFL]). Suppose A = A′′ and let E : C → Abe the vacuum conditional expectation. Then (Tt := E ◦ jt)t≥0 is a semigroup ofcompletely positive maps on A that describes the reduced dynamics of an openquantum system, and j is a dilation of this quantum dynamical semigroup (QDS).

One method of constructing such cocycles is to solve a quantum stochastic dif-ferential equation (QSDE) of Evans-Hudson type ([EvH],[LW1]),

djt = jt ◦ θαβ dΛβ

α(t), j0(a) = a⊗ 1, (0.2)

1991 Mathematics Subject Classification. Primary: 81S25; Secondary: 60H20, 47D06.Key words and phrases. Quantum stochastic, stochastic differential equation, stochastic cocy-

cle, birth and death process, inverse oscillator, diffusion process.

1

2 FRANCO FAGNOLA AND STEPHEN J WILLS

where θ = [θαβ ]α,β≥0 is a matrix of linear maps on A, Λ = [Λβ

α]α,β≥0 are thefundamental noise processes of Hudson-Parthasarathy quantum stochastic calculus([HuP],[Mey],[Par]), and summation over repeated indices is understood. Con-versely it can be shown that all sufficiently well-behaved cocycles arise in thismanner ([AcM],[LW2]). However for applications to physics ([Bar],[Be1],[Sin], andreferences therein), and for the realisation of classical stochastic processes in thequantum setting ([F4],[F5]), it is usually required that the components θα

β of thematrix θ consist of unbounded maps. Proving the existence of a solution to the EHequation is then a highly non-trivial problem ([FSi],[Be2]). An alternative routeexists for the construction of cocycles when A = B(h), the full algebra, by consid-ering the subclass of inner cocycles obtained by conjugation. A family U = (Ut)t≥0

of bounded operators on H := h⊗F is a right cocycle if

U0 = 1; Us+t = σs(Ut)Us, s, t ≥ 0,

and a left cocycle if the adjoint family is a right cocycle. If U is a right cocycle forwhich each Ut is a coisometry then defining j by

jt(X) := U∗t (X ⊗ 1)Ut, X ∈ B(h) (0.3)

produces a cocycle of the form (0.1). As above, right and left operator-valuedcocycles can be constructed by solving the right and left Hudson-Parthasarathyequations:

dUt = Fαβ Ut dΛβ

α(t); dVt = VtGαβ dΛβ

α(t), (0.4)where F = [Fα

β ] and G = [Gαβ ] are matrices of operators on h, and again all

sufficiently regular right and left cocycles arise this way ([F3],[LW2]). Furthermore,if U is the solution to the right equation and each Fα

β is bounded then j definedby (0.3) satisfies the EH equation (0.2) for the matrix of maps θ defined by

θαβ (X) = XFα

β + (F βα )∗X +

∑i≥1(F

iα)∗XF i

β . (0.5)

The equations (0.4) are written in the form usually encountered in the literatureon quantum stochastic calculus, but to give rigorous meaning to these equationsthe operators Fα

β and Gαβ should really be defined as operators on the whole space

H rather than just the first component h of the tensor product. When they arebounded operators they can be identified with Fα

β ⊗ 1 and Gαβ ⊗ 1, the unique con-

tinuous extensions of the algebraic tensor products with the identity operator onFock space, and then no serious difficulty occurs. Similarly, when seeking solutionsto the left equation with unbounded coefficients, since processes are only ever de-fined, in the HP calculus, on the algebraic tensor product of some dense subspaceof h and E (the linear span of the exponential vectors in Fock space) we can iden-tify Gα

β with its algebraic ampliation. To solve the right equation for unboundedFα

β it is clearly necessary to obtain information about the range of the operators(Ut)t≥0. Solutions have been found in [App], [F1], and [Vin], and in all cases thiswas achieved by assuming that there is a dense subspace of

⋂α,β Dom Fα

β consist-ing of vectors satisfying certain analyticity conditions. Subsequently Fagnola ([F3])and Mohari ([Mo1]) focused on the left equation and obtained existence results forthat equation under far less stringent hypotheses on the coefficient matrix since theanalytical difficulties are considerably less.

In this paper we present a new method for solving the right equation that allowsus to incorporate advances made in the study of the left equation. In particularthe domains of the coefficients are no longer required to contain a common denseinvariant subspace. Two advantages of dealing the right equation are that the proofthat the solution is an isometry process is very much easier (cf. Corollary 2.4), andthat the inner cocycle j defined through conjugation by (0.3) is seen to possess aninfinitesimal generator in that it satisfies the EH equation on some ∗-algebra A for

QSDES WITH UNBOUNDED COEFFICIENTS 3

the maps θαβ defined by (0.5). The germ of the idea is as follows: if the coefficients

Gαβ for the left equation are bounded, and the solution V is a contraction process,

then the adjoint process V ∗ is the solution to the adjoint right equation:

dV ∗t = (Gβ

α)∗V ∗t dΛβ

α(t), V ∗0 = 1.

The main result of the paper, Theorem 2.3, extends this principle to the caseof unbounded generators by hypothesising the existence of a positive self-adjoint“reference operator” satisfying a form inequality that can be written heuristicallyas

θ(C) ≤ b(C ⊗ 1).

This enables us to obtain a priori estimates on the continuity of each V ∗t with

respect to the graph norm of C1/2, and hence obtain information about the rangeof V ∗

t . The method was inspired by the techniques developed in [ChF] and [CGQ]for proving the conservativity of QDSs, a problem that has intimate connectionswith proving that solutions to the left equation are isometric (see Proposition 2.5).

The plan of the paper is as follows. Section 1 contains some general resultsabout closable operators and their ampliations, and one-parameter contractionsemigroups. These allow us to define precisely what we mean by a solution ofthe right equation at the start of Section 2, before going onto to establish our mainresult. This is then exploited in Section 3 to give simplified conditions under whichit is possible to construct isometric solutions to the right equation when there isonly one dimension of quantum noise, that is, when k = C. Finally, in Section 4, weapply these results to realise classical birth and death processes as quantum flows,prove the existence and unitarity of a solution to a QSDE that arises in models ofsuperradiation (via an alternative approach to that used in [Wal]), and constructunitary right cocycles that enable us to dilate QDSs of diffusion type (see, for ex-ample, [AlF] and [F4]), as well as realising classical diffusion processes as quantumflows in Fock space.

Tensor product and summation conventions. We shall use the symbol � todenote the algebraic tensor product of vector spaces and linear maps, reserving ⊗for the Hilbert space tensor product of Hilbert spaces and their vectors. If S andT are closable operators on Hilbert spaces h and k respectively, then we denote theclosure of S � T by S ⊗ T (see Lemma 1.1 below). Thus if S ∈ B(h), T ∈ B(k),then S ⊗ T is the unique continuous extension to the Hilbert space h ⊗ k of thebounded operator S � T whose domain is the inner product space h� k. At timeswe will follow the trends prevalent in the literature and identify bounded operatorswith their ampliations, but only when this does not lead to confusion.

We shall adopt the Einstein summation convention and sum over repeated in-dices; greek indices will run from 0 to d, and roman indices from 1 to d, where d isthe number of dimensions of quantum noise (see the start of Section 2).

1. Operator theory preliminaries

In this section we collect together a number of results on closable operators andone-parameter semigroups that we shall need later in the paper.

Lemma 1.1. Let S and T be closable operators on Hilbert spaces h and k respec-tively. Then the operator S � T is closable.

Proof. This follows from the obvious operator inclusion (S � T )∗ ⊃ S∗ � T ∗. �

Remark. Since we denote the closure of S�T by S⊗T , we have that S⊗T = S⊗T .


The main use we make of the above result is to ampliate closable operators, thatis taking T to be the identity. In particular we shall need to consider pairs of closedor closable operators, the domain of one lying inside the domain of the other, andthese behave well under such ampliations.

Lemma 1.2. Let S and T be closable operators on a Hilbert space h such thatDom S ⊂ Dom T . Then Dom S ⊗ 1k ⊂ Dom T ⊗ 1k for every Hilbert space k, andmoreover there exist constants a, b ≥ 0 such that

‖(T ⊗ 1k)ξ‖2 ≤ a‖(S ⊗ 1k)ξ‖2 + b‖ξ‖2 (1.1)

holds for all choices of k and ξ ∈ Dom S ⊗ 1k.

Note. The inequality holds in particular for the case k = C, when S ⊗ 1k = S andT ⊗ 1k = T .

Proof. The inclusion map Dom S ↪→ Dom T is closed and everywhere defined, whenthese spaces are equipped with their respective graph norms, and hence bounded,giving existence of the constants a and b when k = C.

For general k, note that DomS � k is a core for S ⊗ 1k, and that any elementξ of this space can be written as

∑i ui ⊗ vi where ui ∈ Dom S and {vi} is an

orthonormal set. It is then straightforward to check that (1.1) remains valid forsuch ξ and the same a and b, from which the result then follows. �

To define quantum stochastic integrals we work with square-integrable Hilbert-space valued functions, and when dealing with the right HP equation we mustapply closed operators to such functions and determine if the resulting map is againsquare-integrable. The following settles the measurability part of the question.

Lemma 1.3. Let T be a closed operator on a Hilbert space h, and let f : X → h bea strongly measurable function on some measure space X satisfying f(X) ⊂ Dom T .Then the map g : x 7→ Tf(x) is strongly measurable.

Proof. Let T = U |T | be the polar decomposition of T and define maps gn : X → hby gn(x) = U |T |1[0,n](|T |)f(x), where 1[0,n] is the indicator function of [0, n]. Theneach gn is strongly measurable and (gn) converges to g pointwise. �

When applying the lemmas above the operator S will usually be the generatorof a strongly continuous one-parameter semigroup of operators, (Pt)t≥0 say. Then,for any other Hilbert space k, the family of ampliations (Pt ⊗ 1k)t≥1 is a stronglycontinuous one-parameter semigroup. If we denote its generator by S then clearlyS � 1 ⊂ S. But Dom S � k is a dense subspace of h⊗ k that is left invariant by thesemigroup, and thus is a core for S ([Dav], Theorem 1.9). Hence S = S ⊗ 1.

The particular example that we need later is given by taking a positive self-adjoint operator C on h, and letting Q be the contraction semigroup generated by−C. So then (Qt ⊗ 1)t≥0 is generated by −C ⊗ 1. We will make repeated use ofthe following variant of the Yosida approximation:

Cε := RεCRε, where Rε = (1 + εC)−1 for each ε > 0.

The spectral theorem implies that Cε ∈ B(h), and that u ∈ h is in Dom C1/2

if and only if limε↓0 ‖(Cε)1/2u‖ < ∞. Moreover, for any other Hilbert space k,(1h + εC) � 1k is a bijection onto h � k and a restriction of 1h⊗k + εC ⊗ 1k. Thus(1+εC⊗1)−1|h�k = Rε�1k, hence (C⊗1k)ε|h�k = Cε�1k, and so (C⊗1k)ε = Cε⊗1k

by continuity. Thus we can identify Cε with Cε⊗1k in what follows without causingserious harm, since we are actually working with the Yosida approximation of thegenerator of the ampliated semigroup.


The reason for using this variant of the Yosida approximation is that the un-boundedness of the coefficients Fα

β is controlled by multiplying by (Cε)1/2, and sowe need a greater power of C in the denominator than the numerator.

Lemma 1.4. Let C and T be operators on the Hilbert space h, with C positive,invertible and self-adjoint, and T closed. The following are equivalent :

(i) T (Cε)1/2 is densely defined and bounded for all ε > 0;(ii) T (Cε)1/2 is everywhere defined and bounded for all ε > 0;(iii) Dom C1/2 ⊂ Dom T .

Proof. (i ⇒ ii) T (Cε)1/2 is closed since T is closed and (Cε)1/2 is bounded, and sothe result follows by the Closed Graph Theorem.

(ii ⇒ iii) Writing (Cε)1/2 as the product C1/2Rε it is clear that it maps h bijec-tively onto Dom C1/2, which is thus contained in Dom T .

(iii ⇒ i) This follows from Lemma 1.2, since C1/2(Cε)1/2 ∈ B(h). �

Remark. The implications (iii ⇒ i ⇒ ii) remain valid when 0 is in the spectrum ofC. However 0 must be in the resolvent of C for (ii ⇒ iii) — consider C = 0.

2. Fock space and the right and left HP equations

Quantum stochastic integrals. Fix a Hilbert space h, called the initial space,and an integer d ≥ 1, the number of dimensions of quantum noise. Let H = h⊗F ,the Hilbert space tensor product of the initial space and F = Γ(L2(R+; Cd)), thesymmetric Fock space over L2(R+; Cd). Put

M = L2(R+; Cd) ∩ L∞loc(R+; Cd) and E = Lin{ε(f) : f ∈ M},

where ε(f) = ((n!)−1/2f⊗n) is the exponential vector associated to the test functionf . The elementary tensor u⊗ε(f) will usually be abbreviated to uε(f). The notionof adaptedness plays a crucial role in the theory of quantum stochastic calculus asdeveloped by Hudson and Parthasarathy ([HuP]). This is expressed through thecontinuous tensor product factorisation property of Fock space: for each t > 0 let

Ft = Γ(L2([0, t[; Cd)

), F t = Γ

(L2([t,∞[; Cd)

).

Then F = Ft⊗F t via the continuous linear extension of the isometric map ε(f) 7→ε(f |[0,t[)⊗ ε(f |[t,∞[); Ft and F t embed naturally into F as subspaces by tensoringwith the vacuum vector ε(0). Let D be a dense subspace of h. An operator processon D is a family X = (Xt)t≥0 of operators on H satisfying:

(i) D� E ⊂⋂

t≥0 Dom Xt,(ii) t 7→ 〈uε(f), Xtvε(g)〉 is measurable,(iii) Xtvε(g|[0,t[) ∈ h⊗Ft, and Xtvε(g) = [Xtvε(g|[0,t[)]⊗ ε(g|[t,∞[),

for all u ∈ h, v ∈ D, f, g ∈ M and t > 0. Families of operators satisfying (iii) arecalled adapted. Any process satisfying the further condition

(iv) t 7→ Xtvε(g) is strongly measurable and∫ t

0‖Xsvε(g)‖2 ds < ∞ ∀t > 0,

is called stochastically integrable on D. It is for these processes that Hudson andParthasarathy defined the stochastic integral

∫ t

0Xs dΛα

β(s) for each of the funda-mental noise processes Λα

β which are defined with respect to the standard basis ofCd. The resulting family (

∫ t

0Xs dΛα

β(s)) is a process on D, and moreover the mapt 7→

∫ t

0Xs dΛα

β(s) is strongly continuous on D� E . The action of such integrals isgiven in (2.1), and their interaction with each other, the quantum Ito formula, isgiven in (2.2) below. However, rather than give these for a single integral we shallwork with matrices of processes: if M = [Mα

β ]dα,β=0 is a matrix of stochastically in-tegrable processes on D then we can set IM

t =∫ t

0Mα

β (s) dΛβα(s), the sum of (d+1)2


integrals, to produce another (continuous) process on D. Moreover, for all u ∈ h,v ∈ D, f, g ∈ M and t > 0

〈uε(f), IMt vε(g)〉 =

∫ t

0

fα(s)gβ(s)〈uε(f),Mαβ (s)vε(g)〉 ds, (2.1)

where f1, . . . , fd are the components of the Cd-valued function f , f0 ≡ 1, fα(s) =fα(s), and our summation convention is in force. If N = [Nα

β ] is another matrixof stochastically integrable processes on some other dense subspace D′ and we putINt =

∫ t

0Nα

β (s) dΛβα(s) then

〈IMt uε(f), IN

t vε(g)〉 =∫ t

0

fα(s)gβ(s){〈IN

s uε(f), Nαβ (s)vε(g)〉 (2.2)

+ 〈Mβα (s)uε(f), IM

s vε(g)〉+ 〈M iα(s)uε(f), N i

β(s)vε(g)〉}

ds

for all u ∈ D, v ∈ D′, f, g ∈ M and t > 0.Finally, a process X = (Xt)t≥0 on D has a strong stochastic integral represen-

tation if there is a matrix [Mαβ ] of stochastically integrable processes on D such

that

Xt = X0 +∫ t

0

Mαβ (s) dΛβ

α(s).

It follows readily from (2.1) and (2.2) that

‖Xtuε(f)‖2 = ‖X0uε(f)‖2 +∫ t

0

{2 Re〈fα(s)Xsuε(f), fβ(a)Mα

β (s)uε(f)〉

+∑d

i=1‖fα(s)M i

α(s)uε(f)‖2}

ds

(2.3)

for all u ∈ D, f ∈ M and t > 0.

Differential equations. In this paper we are concerned with the right and leftHP equations:

dUt = Fαβ Ut dΛβ

α(t), U0 = 1, (R)

dVt = VtGαβ dΛβ

α(t), V0 = 1, (L)

where F = [Fαβ ]dα,β=0 and G = [Gα

β ]dα,β=0 are matrices of operators on h. Given anysuch matrix F of operators for which each Fα

β is densely defined (respectively clos-able) let F ∗ (resp. F ) denote the matrix [(F β

α )∗] of adjoints (resp. [Fαβ ] of closures).

Associated to any such matrix we define the following subspace of h

Dom[F ] :=⋂

α,β Dom Fαβ . (2.4)

Note that F gives rise naturally to an operator on ⊕(d+1)h by the prescription(uγ) 7→ (Fα

β uβ), which has domain {(uγ) ∈ ⊕(d+1)h : uγ ∈⋂

α Dom Fαγ for each γ}.

The subspace Dom[F ] is the largest subspace D ⊂ h such that ⊕(d+1)D is containedin this maximal domain.

We will only consider solutions that are contraction processes, that is processesU or V for which each Ut or Vt is a contraction. Let D ⊂ h be a dense subspace.A contraction process V is a weak solution of (L) on D for the operator matrix Gif the following hold:

(Li) D ⊂ Dom[G],(Lii) for all u ∈ h, v ∈ D, f, g ∈ M and t > 0,

〈uε(f), (Vt − 1)vε(g)〉 =∫ t

0

fα(s)gβ(s)〈uε(f), VsGαβvε(g)〉 ds. (2.5)


Note that any weak solution is necessarily weakly continuous. The process V is astrong solution of (L) on D for G if, in addition,

(Liii) t 7→ Vtξ is strongly measurable for all ξ ∈ H.The effect of this extra condition is that the processes

(Vt(Gα

β � 1))t≥0

on D arestochastically integrable, since V is assumed to be a contraction process, and sonow by (2.1) and (2.5) it follows that

Vt = 1 +∫ t

0

Vs(Gαβ � 1) dΛβ

α(s).

For the right equation (R) the situation is in general more complex since thereis no reason to expect that for any solution U the range of each Ut should lie inan algebraic tensor product of the form D′ � F . For this reason we only definesolutions of (R) when each component Fα

β of the matrix F is closable. Let F ⊗ 1denote the matrix [Fα

β ⊗ 1] of closed operators on H (so that Dom[F ⊗ 1] ⊂ H), andlet D be as above. A contraction process U is a weak solution of (R) on D for theoperator matrix F if the following hold:

(Ri)⋃

t≥0 Ut(D� E) ⊂ Dom[F ⊗ 1],(Rii) for all u ∈ h, v ∈ D, f, g ∈ M and t > 0

〈uε(f), (Ut − 1)vε(g)〉 =∫ t

0

fα(s)gβ(s)〈uε(f), (Fαβ ⊗ 1)Usvε(g)〉 ds.

Note that since Fαβ is assumed to be closable the measurability of the integrand

follows by taking adjoints and using standard approximation arguments. A strongsolution of (R) on D is any weak solution that satisfies the further condition(Riii) each process (Fα

β ⊗ 1)U on D is stochastically integrable.For such U we have

Ut = 1 +∫ t

0

(Fαβ ⊗ 1)Us dΛβ

α(s),

and so in particular U is strongly continuous.If h is separable then any H-valued weakly measurable function is also strongly

measurable by Pettis’ Theorem, and thus any weak solution to (L) is necessarily astrong solution. However the same need not be true for solutions of (R). A notionof mild solution for (R) has been introduced in [FW]. There it is shown that anystrong solution is also a mild solution, and that there exist coefficient matrices Ffor which mild solutions exist but for which there are no strong solutions.

Let G = [Gαβ ] be a matrix of operators on h, T a positive, self-adjoint operator

on h, and D a subspace of h such that

D ⊂ Dom T ∩Dom[G] and Giβ(D) ⊂ Dom T 1/2 ∀i ≥ 1, β ≥ 0.

Then we can define a real quadratic form θG(T ) and a matrix of sesquilinear forms[θG(T )α

β ] by

θG(T )(u) = 2 Re〈Tuα, Gαβuβ〉+

∑di=1‖T 1/2Gi

βuβ‖2,

and

θG(T )αβ(u, v) = 〈Tu,Gα

βv〉+ 〈Gβαu, Tv〉+ 〈T 1/2Gi

αu, T 1/2Giβv〉

for u = (uα) ∈ ⊕(d+1)D and u, v ∈ D. It follows that

θG(T )(u) = θG(T )αβ(uα, uβ).

We say that θG(T ) is defined as a form on D whenever we need to make the domainof definition precise; if T is bounded then θG(T ) is defined as a form on the subspace


Dom[G] of h. If θG(T ) is in fact bounded then we shall also use θG(T ) to denotethe corresponding bounded self-adjoint operator.

Proposition 2.1 ([F3],[MoP]). Let G = [Gαβ ] be a matrix of operators on h, and

let D be a dense subspace of h. Suppose that there exists a contraction process Vthat is a strong solution to (L) on D for this G. Then θG(1) ≤ 0 as a form on D.If V is an isometry process then θG(1) = 0 on D.

Proof. Let ξ =∑

p upε(fp1[0,T ]) for some T > 0 and some finite family {(up, fp)}in D×M in which each fp is continuous. Then contractivity of V implies that

0 ≥ ‖Vtξ‖2−‖ξ‖2 =∫ t

0

{2 Re〈Vsy

α(s), VsGαβyβ(s)〉+

d∑i=1

‖VsGiαyα(s)‖2

}ds (2.6)

by (2.3), where yα(s) =∑

p fαp (s)upε(fp1[0,T ]). Differentiating at 0 and letting

T → 0 gives0 ≥ θG(1)(y).

where y = (∑

p fαp (0)up). Varying the fp and up then gives the result, and note

that if V is an isometry process then the inequality in (2.6) becomes an equality. �

Remarks. (a) If G is a matrix of operators on h such that the inequality θG(1) ≤ 0holds on some dense subspace D then [δi

j1 + Gij ]

di,j=1 defines a contraction from

⊕(d)D to ⊕(d)h, and so in particular each Gij has a unique continuous extension to

an element of B(h). If θG(1) = 0 then [δij1 + Gi

j ]di,j=1 is an isometry.

(b) If all the components in the matrix G (respectively F ) are bounded then thereis always a unique strong solution V of (L) (resp. a solution U of (R)), althoughit may be an unbounded process on h. In this situation θG(1) ≤ 0 is not onlya necessary condition for contractivity of V but also sufficient one. Similarly Uwill be a contraction process if and only if θF (1) ≤ 0. The original proofs of thischaracterisation are contained in [F3] and [Mo2]; an alternative line of proof isgiven in [LiP] and [LW1] that makes use of the characterisation of the generatorsof completely positive contraction flows. In this context it makes sense to regardθF as the linear map B(h) → Md+1(B(h)) given by

θF (X) = (X ⊗ 1)F + F ∗(X ⊗ 1) + F ∗∆(X)F,

where ∆(X) = diag{0, X, . . . , X}, rather than just restricting it to the cone ofpositive self-adjoint operators.

Taking adjoints. Suppose that V is a contraction process that is a weak solutionto (L) on D for some operator matrix G, and also that each Gα

β ∈ B(h). Then itfollows from (2.1) that V ∗ is a weak solution to the QSDE dV ∗ = (Gβ

α)∗V ∗ dΛβα on

h. Our main result of the section shows how to extend this procedure to a classof generators G for which the Gα

β are no longer bounded. In particular we mustobtain information about the range of each V ∗

t . Our arguments make use of thequantum Ito formula (2.2), which is valid for processes that have strong stochasticintegral representations, and thus our standing hypothesis is the existence of astrong solution to (L), from which we will prove the existence of a strong solutionto (R). In Section 3 we give conditions that guarantee the existence of this solutionto (L).

As part of the proof we will require that the adjoint process V ∗ be stronglymeasurable, and Proposition 2.2 below gives some sufficient conditions for this to bethe case. In fact we shall show that it is strongly right continuous by first showing itis a Markovian cocycle and then adapting standard arguments of semigroup theory.For each t ∈ R let st be the unitary right shift operator on L2(R; Cd), defined by


(stf)(r) = f(r − t) for f ∈ L2(R; Cd). Let St be the second quantisation andampliation of st, that is Stuε(f) = uε(stf). Then the map

B(h⊗ Γ(L2(R; Cd))

)3 Y 7→ StY S∗t ∈ B

(h⊗ Γ(L2(R; Cd))

)is a normal automorphism, and the collection of these for all t ∈ R is an ultraweaklycontinuous one-parameter group of such maps. Now let X ∈ B(H), then ampliatingwith 1−, the identity of Γ(L2(]−∞, 0[; Cd)), we get X ⊗ 1− ∈ B

(h⊗Γ(L2(R; Cd))

).

If t ≥ 0 it follows that there is some σt(X) ∈ B(H) such that

σt(X)⊗ 1− = St(X ⊗ 1−)S∗t .

The family (σt)t≥0 so defined is an ultraweakly continuous one-parameter semigroupof unital, normal ∗-homomorphisms of B(H). A family W = (Wt)t≥0 ⊂ B(H) is aleft cocycle if it satisfies the following:

(i) The family W is adapted.(ii) W0 = 1.(iii) Ws+t = Wsσs(Wt) for all s, t ≥ 0.

Similarly, W is a right cocycle if W ∗ = (W ∗t )t≥0 is a left cocycle.

Proposition 2.2. Let G be a matrix of operators on h, and suppose that there existsa contraction process V that is a strong solution to (L) on some dense subspaceD ⊂ h . If V is the unique strong solution for this G and D then V is stronglycontinuous and a left cocycle. Furthermore V ∗ is a right cocycle that is stronglyright continuous.

Proof. That V is strongly continuous is a consequence of its strong stochastic inte-gral representation as noted earlier. So now fix t > 0 and consider the process V t

defined by

V ts =

{Vs, s ≤ t

Vtσt(Vs−t), s > t.

It follows that V t is a strong solution to (L) on D� E for G, and so by uniquenessV is a left cocycle. Thus V ∗ is a right cocycle by definition.

Now for any s, t ≥ 0 and ξ ∈ H we have

‖(V ∗s+t − V ∗

t )ξ‖2 = ‖(σt(V ∗s )− 1)V ∗

t ξ‖2

≤ 2‖V ∗t ξ‖2 − 2 Re〈V ∗

t ξ, σt(Vs)V ∗t ξ〉

since σt(V ∗s ) is a contraction. The right hand side converges to zero as s → 0 by

strong continuity of V and normality of σt, and the result follows. �

Remark. Mohari proved the following uniqueness result in [Mo1]: let G = [Gαβ ]

be an operator matrix with G00 the generator of a strongly continuous contraction

semigroup. If D ⊂ Dom[G] is a core for G00 then there is at most one weak solution

V to (L) on D for this G. In fact in [Mo1] it is assumed from the outset thath is separable and so there is no distinction between weak and strong solutions.However, using this uniqueness result, the arguments of the proof above can beadapted to show that if a weak solution to (L) does exist then it is a cocycle, henceit is strongly continuous, and so it must actually be a strong solution.

Given any positive self-adjoint operator T on h let ι(T ) denote the form u 7→∑dα=0 ‖T 1/2uα‖2, defined for each u ∈ ⊕(d+1) Dom T 1/2. Also, recall the notation

Dom[F ] introduced in (2.4)

Theorem 2.3. Suppose that U is a contraction process, F is an operator matrix,C is a positive self-adjoint operator on h, and δ > 0 and b1, b2 ≥ 0 are constantssuch that the following hold :


(i) There is a dense subspace D ⊂ h such that the adjoint process U∗ is a strongsolution of dU∗ = U∗(F β

α )∗ dΛβα on D, and is the unique strong solution for

this F ∗ = [(F βα )∗] and D.

(ii) For each 0 < ε < δ there is a dense subspace Dε ⊂ D such that (Cε)1/2(Dε) ⊂D and each (Fα

β )∗(Cε)1/2|Dε is bounded.(iii) Dom C1/2 ⊂ Dom[F ].(iv) Dom[F ] is dense in h, and for all 0 < ε < δ the form θF (Cε) on Dom[F ]

satisfies the inequality

θF (Cε) ≤ b1ι(Cε) + b21

on some dense subspace of Dom[F ].Then U is a strong solution to the right equation (R) on Dom C1/2 for the operatormatrix F .

Note. By (i) it follows that each Fαβ is closable, hence the matrix F is defined,

and so (iii) makes sense.

Proof. First note that (Fαβ )∗(Cε)1/2 is bounded and everywhere defined by (ii) and

Lemma 1.4. Taking adjoints we have

B(h) 3 [(Fαβ )∗(Cε)1/2]∗ ⊃ (Cε)1/2Fα

β ⊃ (Cε)1/2Fαβ ,

Thus the form θF (Cε) on Dom[F ] is defined in terms of bounded operators thathave a dense common domain of definition. Using the operators [(Fα

β )∗(Cε)1/2]∗

we can define an extension of θF (Cε) to a bounded form on all of h, and so we shalltreat it as a bounded operator on ⊕(d+1)h, also identifying it with its ampliation toan operator on ⊕(d+1)H. Moreover the inequality in (iv) is now valid as an operatorinequality.

Now by (i) the process U∗ satisfies

U∗t = 1 +

∫ t

0

U∗s (F β

α )∗ dΛβα(s)

on the domain D � E , and so it follows that the process (U∗t (Cε)1/2)t≥0 has the

stochastic integral representation

U∗t (Cε)1/2 = (Cε)1/2 +

∫ t

0

U∗s (F β

α )∗(Cε)1/2 dΛβα(s)

on Dε�E , which extends to all of h�E by continuity. By (i) and Proposition 2.2 itfollows that U∗ is a strongly continuous left cocycle and so U is a strongly (right)continuous right cocycle. Thus we can take the adjoint of the above, since theresulting integrands are stochastically integrable, to get

(Cε)1/2Ut = (Cε)1/2 +∫ t

0

[(Fαβ )∗(Cε)1/2]∗Us dΛβ

α(s)

on h� E . Applying (2.3) gives

‖(Cε)1/2Utuε(f)‖2 = ‖(Cε)1/2uε(f)‖2

+∫ t

0

{2 Re〈fα(s)(Cε)1/2Usuε(f), fβ(s)[(Fα

β )∗(Cε)1/2]∗Usuε(f)〉

+∑d

i=1‖fα(s)[(F iα)∗(Cε)1/2]∗Usuε(f)‖2

}ds.

for all u ∈ h, f ∈ M. Collecting together the terms making up θF (Cε) we have

‖(Cε)1/2Utuε(f)‖2 = ‖(Cε)1/2uε(f)‖2 +∫ t

0

〈x(s), θF (Cε)x(s)〉 ds


where xα(s) := fα(s)Usuε(f). The inequality in (iv) implies that

‖(Cε)1/2Utuε(f)‖2 ≤ ‖(Cε)1/2uε(f)‖2

+∫ t

0

(b1‖(Cε)1/2Usuε(f)‖2 + b2‖Usuε(f)‖2

)dνf (s)

where νf (t) =∫ t

0(1 + ‖f(s)‖2) ds, and so the Gronwall inequality gives

‖(Cε)1/2Utuε(f)‖2 ≤{‖(Cε)1/2uε(f)‖2 + b2

∫ t

0

‖Usuε(f)‖2 dνf (s)}

exp{b1νf (t)}.

Letting ε → 0 we see that Ut

((Dom C1/2

)�E) ⊂ Dom C1/2⊗ 1, and condition (iii)

in conjunction with Lemma 1.2 implies that Dom C1/2 ⊗ 1 ⊂ Dom Fαβ ⊗ 1 for all

α, β, and hence U is a weak solution of (R) on Dom C1/2.Now by Lemma 1.3 the functions t 7→ (Fα

β ⊗ 1)Utuε(f) are strongly measurablefor all 0 ≤ α, β ≤ d, u ∈ Dom C1/2 and f ∈ M. Also the above inequality (in thelimit as ε → 0) shows that t 7→ ‖(C1/2 ⊗ 1)Utuε(f)‖ is locally bounded, and soLemma 1.2 and (iii) imply that the processes

((Fα

β ⊗ 1)Ut

)t≥0

are stochasticallyintegrable on Dom C1/2. Hence U is a strong solution as required. �

Remarks. (a) The requirement that U∗ be the unique strong solution to the ad-joint left equation allowed us to conclude that the processes

{[(Fα

β )∗(Cε)1/2]∗U}

are stochastically integrable and hence (Cε)1/2U has a strong stochastic integralrepresentation. If h is separable then the stochastic integrability of this family ofprocesses is guaranteed by the equivalence of strong and weak measurability forfunctions taking values in a separable Hilbert space, and so the uniqueness require-ment in part (i) of the hypothesis can be dropped without affecting the result.

(b) By Lemma 1.4 a sufficient condition for the boundedness of each (F βα )∗(Cε)|Dε

is Dom C1/2 ⊂ Dom[F ∗], and indeed this is necessary if 0 lies in the resolvent ofC. This observation provides an important guide as to what would be a suitablechoice for C. As an illustration, in the diffusion example in Section 4 we have that(F 0

0 )∗ is a second order differential operator, and so for C we take ∂4 + 1. Howeverit is still important to check that (Cε)1/2 maps some dense subspace Dε into D, thesubspace for which U∗ satisfies the QSDE (L).

(c) The proof of the above result remains valid if we replace Cε by other variantsof the Yosida approximation which raises the possibility of using a “less unbounded”reference operator C. Indeed we could use CR4

ε or C(1+ εC2)−2 instead of Cε, andthen we would be able to use C of the same order as (F 0

0 )∗. However if we adoptthese variants then proving the analogous result to Proposition 3.4 below becomesmuch harder.

Having constructed a solution U to the right equation we now finish this sectionwith two results that give conditions under which we can prove that U is an isometryor coisometry process. The first of these happens, at least formally, when θF (1) = 0(cf. the remark after Proposition 2.1), although we must take care when consideringthe form θF (1), defined on Dom[F ], and its extension θF (1) to Dom[F ].

Corollary 2.4. Suppose that the conditions of Theorem 2.3 hold and let U be thestrong solution to (R) for the given matrix F . If either

(i) Dom C1/2 ∩Dom[F ] is a core for C1/2 and θF (1) = 0, or(ii) θF (1) = 0,

then U is an isometry process .

Proof. If (i) holds then, using the condition (iii) from the hypotheses of Theorem 2.3and Lemma 1.2, it is possible to find for each u ∈ Dom C1/2 a sequence (un) in


this core such that un → u and Fαβ un → Fα

β u. It follows that the form θF (1) whenrestricted to Dom C1/2 is equal to zero, which is also clearly the case if condition (ii)holds. So now let ΘF denote the sesquilinear form defined by(

(ξγ), (ηγ))7→ 〈ξα, (Fα

β ⊗ 1)ηβ〉+ 〈(F βα ⊗ 1)ξα, ηβ〉+ 〈(F i

α ⊗ 1)ξα, (F iβ ⊗ 1)ηβ〉

for (ξγ), (ηγ) ∈ ⊕(d+1) Dom[F ⊗ 1]. By the above the restriction of this form toDom C1/2 � F is zero. But as noted in Section 1, Dom C1/2 � F is a core forC1/2 ⊗ 1, and so another application of the inequality (1.1) allows us to show thatΘF is zero when restricted to DomC1/2⊗1. Now since U is a strong solution to (R)we can apply (2.2) to get

〈Utuε(f), Utvε(g)〉 = 〈uε(f), vε(g)〉+∫ t

0

ΘF

(x(s), y(s)

)ds

for all u, v ∈ Dom C1/2, f, g ∈ M, and where xα(s) = fα(s)Usuε(f) and yα(s) =gα(s)Usvε(g). But Ut maps Dom C1/2�E into Dom C1/2⊗1 by Theorem 2.3, hencethe integrand is zero if either (i) or (ii) holds, and the result follows. �

A quantum dynamical semigroup (QDS) on B(h) is an ultraweakly continuoussemigroup T = (Tt)t≥0 of normal completely positive maps on B(h). It is conser-vative if Tt(1) = 1 for all t ≥ 0. Given the generator K of a strongly continuouscontraction semigroup and operators (Ll)l≥0 such that

〈u, Ku〉+ 〈Ku, u〉+∑

l≥1‖Llu‖2 ≤ 0, (2.7)

for all u in some core D for K, it is possible to construct the minimal QDS T (see[F5] and references therein) that has (formal) generator L given by

〈u,L(X)v〉 = 〈u, XKv〉+ 〈Ku, Xv〉+∑

l≥1〈Llu, XLlv〉, u, v ∈ D.

That is, T is a QDS that satisfies

〈u, Tt(X)v〉 = 〈u, Xv〉+∫ t

0

〈u,L(Ts(X))v〉 ds,

and if T ′ is another QDS satisfying the above integral identity then Tt(X) ≤ T ′t (X)for all t ≥ 0 and positive X ∈ B(h). Showing that the solution U to (R) constructedin Theorem 2.3 is a coisometry process is equivalent to showing that U∗ is anisometry process, which (under favourable circumstances) is equivalent to showingthat a related QDS is conservative:

Proposition 2.5 ([F3],[F5]). Suppose that the conditions of Theorem 2.3 holdand let U be the strong solution to (R) for the given matrix F . Suppose furtherthat (F 0

0 )∗ is the generator of a strongly continuous contraction semigroup, that thesubspace D is a core for (F 0

0 )∗, and let T be the minimal QDS with generator

〈u,L(X)v〉 = 〈u, X(F 00 )∗v〉+ 〈(F 0

0 )∗u, Xv〉+∑d

i=1〈(F 0i )∗u, X(F 0

i )∗v〉.

The following are equivalent :

(i) U is a coisometry process.(ii) θF∗(1) = 0 on D and T is conservative.(iii) [δi

j1 + F ij ]

di,j=1 is a coisometry on ⊕d

i=1h and T is conservative.

Remark. By Proposition 2.1 the inequality (2.7) holds for K = (F 00 )∗ and Ll =

(F 0l )∗, hence the minimal QDS T exists.


3. Special case: Isometric solutions with one dimensional noise

The results in the previous section provide a very general method for generatingcontraction solutions to the right HP equation, and one whose basic idea could bemodified easily if necessary, for instance by using different regularisations to Cε.In this section we refine our basic result in a number of ways. Firstly in order tomake use of known results on the existence of (strong) solutions to the QSDE (L)(and hence verify part (i) of Theorem 2.3) we shall assume from now on that theinitial space h is separable. Secondly in order to simplify the form of the generatorwe shall set d = 1, that is we work with only one dimension of quantum noise, andwe now look for isometric solutions to the QSDE (R) (cf. Proposition 2.1).

The operator matrix for the rest of the section is specified by a triple of operators(L, H, S), where L and H are densely defined, with L closable and H symmetric,and S is an isometric element of B(h). The operator matrix is

F =[− 1

2L∗L− iH −L∗SL S − 1

],

and we assume throughout that Dom[F ] is a dense subspace of h. Rather thanwork with F ∗, the matrix of adjoints, we shall use the following matrix:

F † =[− 1

2L∗L + iH L∗

−S∗L S∗ − 1

],

whose components are restrictions of the components of F ∗. Thus the QSDEs thatwe are now working with are

dUt =[(− 1

2L∗L− iH) dt− L∗S dAt + LdA†t + (S − 1) dΛt

]Ut (R)′

and

dU∗t = U∗

t

[(− 1

2L∗L + iH) dt + L∗ dAt − S∗LdA†t + (S∗ − 1) dΛt

]. (L)′

The formal generator of the related flow is

θF (X) =[− 1

2XL∗L + L∗XL− 12L∗LX + i[H,X] [L∗, X]S

S∗[X, L] S∗XS −X

], (3.1)

so in particular θF (1) = 0 on some domain.

Proposition 3.1. Let (L, H, S) be a triple of operators as above, and suppose thatthere is a dense subspace D of h such that

(i) D ⊂ Dom L∗L ∩Dom L∗ ∩Dom H, and(ii) the closure of (− 1

2L∗L + iH)|D is the infinitesimal generator of a stronglycontinuous contraction semigroup on h.

Then there is a contraction process U∗ that is a strong solution to (L)′ on D, andfurthermore it is the unique strong solution.

Proof. The result follows immediately from the method given in [F3]. The form

θF †(1) : u 7→ 2 Re〈u, F †u〉+ 〈F †u,∆(1)F †u〉 (3.2)

is well-defined for u ∈ D ⊕D by (i), where ∆(1) =[

0 00 1

]. By the construction of

F † we haveθF †(1)(u) = 〈Lu0 − u1, (SS∗ − 1)(Lu0 − u1)〉 ≤ 0 (3.3)

for all u = [u0, u1]> ∈ D⊕D, that is F † satisfies the formal contractivity conditions(cf. the remark after Proposition 2.1). Let K denote the closure of (− 1

2L∗L +iH)|D, then by considering vectors of the form u = [u0, 0]> in (3.2) we can extend(−S∗L)|D to all of Dom K by approximating elements of Dom K by sequencesin D that converge in graph norm and using the inequality (3.3). Thus θF †(1)


extends to Dom K ⊕D, and continues to satisfies (3.3) on this domain. Put In =diag{n(n−K)−1, 1} and F †

n = I∗nF †In for each n ≥ 1, then it follows that F †n is a

bounded map satisfying

F †n + (F †

n)∗ + (F †n)∗∆(1)F †

n ≤ 0.

So for each n we can solve the equation

dU (n)∗ = U (n)∗(F †n)α

β dΛβα,

and each U (n)∗ is contractive. In fact for each u ∈ h, f ∈ M and ξ ∈ H the family{〈ξ, U (n)∗

t uε(f)〉}∞n=1 is equibounded and equicontinuous on each bounded interval.A diagonalisation argument and the Ascoli-Arzela theorem can then be employedto show that there is a weakly convergent subsequence {U (nk)∗} whose limit is therequired solution. The uniqueness follows by the result of Mohari. �

Remark. In the examples below we shall always take S to be unitary since we willbe constructing unitary cocycles. Thus S∗L is closable with S∗L = S∗L. Also,from (3.3), it follows that Dom L ⊃ Dom K, and that the extension of (−S∗L)|Dto Dom K in the proof is nothing but the restriction of −S∗L to this domain.

The next result gives some sufficient conditions that imply that the solution U∗

to (L) constructed above is an isometry process. The conditions are by no meansoptimal, in particular they are not necessary, but are written in such a way as tobe easily applicable to our examples in the next section.

Corollary 3.2. Let (L,H, S) be a triple of operators as above, and suppose thatthe conditions of Proposition 3.1 hold, with U∗ the contraction process that is thestrong solution to (L)′. Suppose further that the operator S is unitary, and thatthere is a positive self-adjoint operator M on h and a constant k ≥ 0 such that :

(i) Dom K ⊂ Dom M1/2, and Dom K is a core for M1/2,(ii) S∗L(Dom K2) ⊂ Dom M1/2,(iii) 2 Re〈M1/2u, M1/2Ku〉+ ‖M1/2S∗Lu‖2 ≤ k‖M1/2u‖2 ∀u ∈ Dom K2, and(iv) Dom M ⊂ Dom L∗L, and ‖Lu‖ ≤ ‖M1/2u‖ ∀u ∈ Dom M ,

where K denotes the closure of (− 12L∗L+ iH)|D. Then U∗ is an isometric process.

Proof. We can apply Theorem 4.4 of [ChF] to deduce that the minimal QDS associ-ated to (L)′ is conservative, and so U∗ is an isometry process by Proposition 2.5. �

Perhaps one of the more difficult things to check in order to be able to applyTheorem 2.3 is that the form inequality for θF (Cε) in (iv) holds for all values ofε in an interval of the form (0, δ). Proposition 3.4 below shows that this will bethe case if the analogous form inequality holds for θF (C), and if the commutatorsof C with L and S are sufficiently well-behaved. The following lemma eases thealgebraic burden of the proof of this result.

Lemma 3.3. Let B be a unital associative algebra and suppose that c, r ∈ B satisfyεcr = εrc = 1 − r for some ε > 0. For any a, b ∈ B the linear maps τa,b and ωa,b

on B defined by

τa,b(x) = a[x, b] and ωa,b(x) = [a, x]b (x ∈ B)

satisfy

τa,b(rcr) = r2τa,b(c)r2 − r(1− r)τa,b(c)(1− r)r − εr[c, a]r2[b, c]r

+ εr(1− r)[c, a]r[b, c](1− r)r − εr2[c, a]r[b, c]r2


and

ωa,b(rcr) = r2ωa,b(c)r2 − r(1− r)ωa,b(c)(1− r)r − εr[c, a]r2[b, c]r

+ εr(1− r)[c, a]r[b, c](1− r)r − εr2[c, a]r[b, c]r2.

Proof. The relations satisfied by c and r imply that [x, r] = εr[c, x]r and hence[x, r2] = εr2[c, x]r + εr[c, x]r2 for all x ∈ B. These identities and those alreadygiven lead to the following chain of equalities:

τa,b(rcr) = a[rcr, b] = arc[r, b] + ar[c, b]r + a[r, b]cr

= ar(1− r)[b, c]r + ar[c, b]r + ar[b, c](1− r)r

= −r2a[b, c]r − [a, r2][b, c]r + ra[b, c](1− r)r + [a, r][b, c](1− r)r

= r2τa,b(c)r − εr2[c, a]r[b, c]r − εr[c, a]r2[b, c]r

+ rτa,b(c)(r − 1)r + εr[c, a]r[b, c](1− r)r

= r2τa,b(c)r2 − r(1− r)τa,b(c)(1− r)r − εr[c, a]r2[b, c]r

+ εr(1− r)[c, a]r[b, c](1− r)r − εr2[c, a]r[b, c]r2,

giving the identity for τa,b. The one for ωa,b follows by an almost identical proof. �

Remark. If B is assumed to be involutive and c and r are self-adjoint, then theidentity for ω can be derived from that for τ since ωa,b(x) = τb∗,a∗(x∗)∗.

Proposition 3.4. Let (L,H, S) be a triple of operators as above, and suppose thatthere exists a positive, invertible, self-adjoint operator C, a dense subspace D of h,and constants 0 < δ < 1 and b3, b4 ≥ 0 such that the following hold :

(i) Rε(D) ⊂ D for all 0 < ε < δ, and D is contained in the domain of thefollowing operators:

L∗L,L∗S, [C,L], [C,H], CS.

(ii) The form θF (C) defined on D satisfies the following inequality :

−b3ι(C) ≤ θF (C) ≤ b3ι(C).

(iii) For all u ∈ D, ‖C−1/2[C,L]u‖ ≤ b4‖C1/2u‖.(iv) For all u ∈ D, ‖[C,S]u‖ ≤ b4‖C1/2u‖.

Then for all 0 < ε < δ the form θF (Cε) is well-defined on D and satisfies

θF (Cε) ≤ 2(b3 + b4)ι(Cε).

Proof. We shall use the preceding lemma to rewrite each component θαβ (Cε) :=

θF (Cε)αβ of the form (3.1). This is possible, since in the notation of the lemma (and

ignoring domain problems for now) we have

θ00(X) = L∗[ 12L,X] + [ 12L∗, X]L + [iH, X]

= τL∗, 12 L(X) + ω 1

2 L∗,L(X) + ωiH,1(X)

and, similarly,

θ01 = ωL∗,S , θ1

0 = τS∗,L, θ11 = τS∗, 1

2 S + ω 12 S∗,S ,

noting for θ11 that S∗S = 1. Now note that the issue of domains is covered for us

by condition (i). Indeed, each component θαβ (Cε) is a well-defined sesquilinear form

on D, and moreover so is each of the terms such as τL∗, 12 L(Cε). For example we

haveθ01(Cε) = ωL∗,S(RεCRε) = [L∗, RεCRε]S


which should really be thought of as the form

D× D 3 (u, v) 7→ 〈Lu,RεCRεSv〉 − 〈RεCRεu, L∗Sv〉,

which is well-defined since CRε is bounded and D ⊂ Dom L∩Dom L∗S. Lemma 3.3allows us to rewrite this as

θ01(Cε) = R2

εθ01(C)R2

ε −Rε(1−Rε)θ01(C)(1−Rε)Rε

− εRε[C,L∗]R2ε [S, C]Rε + εRε(1−Rε)[C,L∗]Rε[S, C](1−Rε)Rε

− εR2ε [C,L∗]Rε[S, C]R2

ε

and again each of the terms on the right hand side, plus those appearing in thederivation of the above, make good sense courtesy of condition (i). The adjointidentity holds for θ1

0(Cε),

θ00(Cε) = R2

εθ00(C)R2

ε −Rε(1−Rε)θ00(C)(1−Rε)Rε

− εRε[C,L∗]R2ε [L,C]Rε + εRε(1−Rε)[C,L∗]Rε[L,C](1−Rε)Rε

− εR2ε [C,L∗]Rε[L,C]R2

ε ,

and the identity for θ11(Cε) is got by changing θ0

0 to θ11 and L to S in the above.

Thus each component θαβ (Cε) defines a sesquilinear form on D, and the quadratic

form θF (Cε) is well-defined on D with

θF (Cε) = R1εθF (C)R1

ε −R2εθF (C)R2

ε − εR1εφε(C)R1

ε + εR2εφε(C)R2

ε

− ε

[Rε[C,L∗]R2

ε [L, C]Rε Rε[C,L∗]R2ε [S, C]Rε

Rε[C,S∗]R2ε [L,C]Rε Rε[C,S∗]R2

ε [S, C]Rε

] (3.4)

where

R1ε =

[R2

ε 00 R2

ε

], R2

ε =[Rε(1−Rε) 0

0 Rε(1−Rε)

], and

φε(C) =[[C,L∗]Rε[L,C] [C,L∗]Rε[S, C][C,S∗]Rε[L, C] [C,S∗]Rε[S, C]

]are all positive operator matrices or forms.

Now Rε and 1 − Rε are positive contractions, and εRε ≤ C−1 for each ε > 0.Thus for any u = [u0, u1]> ∈ D⊕ D and 0 < ε < δ < 1,

φε(C)(u) = ‖R1/2ε [L,C]u0‖2 + 2 Re〈R1/2

ε [C,L]u0, R1/2ε [S, C]u1〉+ ‖R1/2

ε [S, C]u1‖2

≤ 2{‖R1/2

ε [C,L]u0‖2 + ‖[C,S]u1‖2}

≤ 2ε−1{‖C−1/2[C,L]u‖2 + ‖[C,S]v‖2

}≤ 2ε−1 b4ι(C)(u),

by inequalities (iii) and (iv). Thus the last three terms of (3.4) are bounded above by2b4R2

ε ι(C)R2ε . The result now follows since for all 0 < ε < δ we have R2

εCR2ε ≤ Cε

and Rε(1−Rε)C(1−Rε)Rε ≤ Cε. �

Theorem 3.5. Let (L,H, S) be a triple of operators as above and suppose thatthere is a positive, invertible, self-adjoint operator C such that the hypotheses ofPropositions 3.1 and 3.4 hold. Suppose also that the following conditions hold forsome 0 < δ′ < δ:

(i) For each 0 < ε < δ′ there is a dense subspace Dε ⊂ D such that (Cε)1/2(Dε) ⊂D, and such that the restrictions of the operators L∗(Cε)1/2, L(Cε)1/2 and(− 1

2L∗L + iH)(Cε)1/2 to Dε are bounded.(ii) Dom C1/2 ⊂ Dom− 1

2L∗L− iH ∩Dom L ∩Dom L∗S.


Then U , the adjoint of the process from Proposition 3.1, is a strong solution to (R)′

on Dom C1/2. Moreover if Dom C1/2 ∩ Dom[F ] is a core for C1/2 then U is iso-metric.

Proof. Since the hypotheses of Propositions 3.1 and 3.4 and the additional condi-tions above hold it follows that all of the requirements of Theorem 2.3 are fulfilled,and so the contraction process U is a strong solution to (R)′ for the operator matrixF on the domain Dom C1/2. If Dom C1/2 ∩Dom[F ] is a core for C1/2 then Corol-lary 2.4 can be applied to show that U is an isometry process since θF (1) = 0. �

4. Examples

Birth and death processes. Let h = l2(Z), with standard orthonormal basis(en)n∈Z, and let W be the unitary right shift given by Wen = en+1. Let N be thenumber operator on l2(Z), that is

Dom N = {(un) : un ∈ C,∑

n∈Zn2|un|2 < ∞}, N(un) = (nun).

Then for any function λ : Z → C the operator denoted λ(N) is defined by

Dom λ(N) = {(un) : un ∈ C,∑

n∈Z|λ(n)un|2 < ∞}, λ(N)(un) = (λ(n)un).

Let D0 = Lin{en}, so then D0 ⊂ Dom λ(N) for any function λ, and is in fact acore for λ(N).

Define the triple (L,H, S) and the reference operator C by

L = Wλ(N), H = 0, S = W, C = N2 + 1,

and note that D0 is an invariant subspace for all these.Now L∗L = |λ|2(N), a positive self-adjoint operator for which D0 is a core, so

we can apply Proposition 3.1 and Corollary 3.2 (with D = D0,M = |λ|2(N), andk = 0) to obtain an isometric process U∗ that is a strong solution to (L)′ on D0.

For any function ϕ : Z → C, the form θ(ϕ(N)) is well-defined on D0. Indeed,it is actually possible to regard θF (ϕ(N)) as an operator on h ⊕ h with domainD0 ⊕D0, and it can be written

θF (ϕ(N)) =[λ(N)

1

](ϕ(N + 1)− ϕ(N))

[λ(N) 1

].

We now restrict our attention to functions λ for which there is some b > 0 suchthat

|λ(n)|2 ≤ b(|n|+ 1) ∀n ∈ Z.

Since−(2|N |+ 1) ≤ [(N + 1)2 + 1]− [N2 + 1] = 2N + 1 ≤ 2|N |+ 1,

it follows readily that the form θF (C) satisfies the inequality in part (ii) of Propo-sition 3.4 on the domain D0. Also, note that on D0

[C,L] = [N2,W ]λ(N) = W (W ∗N2W −N2)λ(N) = W (2N + 1)λ(N),

which is relatively bounded by C3/4, so that (iii) follows, and similarly

[C,S] = W (W ∗N2W −N2) = W (2N + 1),

from which part (iv) follows. Thus we can apply Proposition 3.4 to get that θF (Cε)satisfies a form inequality of the correct type on D := D0.

Finally, the growth condition on λ implies that DomC1/2 is contained in thesubspace Dom |λ|2(N) ∩ Dom λ(N) ∩ Dom λ(N)W ∗, so parts (i) and (ii) of Theo-rem 3.5 are satisfied with Dε = D0. Thus there is a strong unitary solution to theQSDE

dUt =[− 1

2 |λ|2(N) dt− λ(N) dAt + Wλ(N) dA†t + (W − 1) dΛt

]Ut


on Dom C1/2, since D0 ⊂ Dom[F ], and D0 is a core for C1/2.If we consider the algebra l∞(Z) ⊂ B(h) acting by pointwise multiplication, then

θ00(ϕ(N)) = |λ|2(N){ϕ(N + 1)− ϕ(N)}, ϕ ∈ l∞(N),

and so the flow X 7→ U∗(X ⊗ 1)U gives a realisation of the classical pure birthprocess with intensity |λ|2, since the generator θ0

0 is of the appropriate form.Replacing the triple above by

L = W ∗µ(N), H = 0, S = W ∗,

with µ : Z → C subject to the same growth condition, and setting C = N2 +1 oncemore, we obtain a strong unitary solution to the QSDE

dUt =[− 1

2 |µ|2(N) dt− µ(N) dAt + W ∗µ(N) dA†t + (W ∗ − 1) dΛt

]Ut,

on Dom C1/2. This time

θ00(ϕ(N)) = |µ|2(N){ϕ(N − 1)− ϕ(N)},

and so we have a realisation of the classical pure death process with intensity µ.By increasing the number of noise dimensions to two we are able to realise a

combined birth and death process. Let F be the operator matrix

F =

− 12 |λ|

2(N)− 12 |µ|

2(N) −λ(N) −µ(N)Wλ(N) W − 1 0W ∗µ(N) 0 W ∗ − 1

.

Then for any function ϕ : Z → C the operator matrix θF (ϕ(N)) (with domaincontaining D0 ⊕D0 ⊕D0) decomposes as|λ|2(N) λ(N) 0

λ(N) 1 00 0 0

ι(+) +

|µ|2(N) 0 µ(N)0 0 0

µ(N) 0 1

ι(−). (4.1)

where ι(+) = ι(ϕ(N +1)−ϕ(N)) and ι(−) = ι(ϕ(N−1)−ϕ(N)). Viewing each com-ponent as a 2× 2 matrix, the estimates above in conjunction with Proposition 3.4shows that there is some b′ > 0 such that θF (Cε) ≤ b′ι(Cε), where C = N2 + 1as before. So now appealing directly to Theorem 2.3 (rather than Theorem 3.5),Corollary 2.4 and Proposition 2.5 (or, rather, a two dimensional version of Corol-lary 3.2), we can show that there is a unitary process U that is a strong solutionto

dUt =[− 1

2 (|λ|2(N) + |µ|2(N)) dΛ00(t)− λ(N) dΛ0

1(t)− µ(N) dΛ02(t)

+ Wλ(N) dΛ10(t) + W ∗µ(N) dΛ2

0(t) + (W − 1) dΛ11(t) + (W ∗ − 1) dΛ2

2(t)]Ut.

Then note from (4.1) that θ00(ϕ(N)) has the required form for the cocycle X 7→

U∗t (X ⊗ 1)Ut to be a realisation of the birth and death process with intensities λ

and µ.

The inverse harmonic oscillator. Let h = l2(Z+), where Z+ = {0, 1, 2, . . .},equipped with the standard basis (en)n≥0. Let W be the isometric right shift,Wen = en+1, and D0 = Lin{en}. Define the triple (L,H, S) and reference operatorC by

L = λ(N)W, H = µ(N), S = 1, C = N2 + 1

for functions λ : Z+ → C and µ : Z+ → R, and assume that there is some c > 0such that

max{|λ(n)|2, |µ(n)|} ≤ c(n + 1) ∀n ≥ 0.


So now for any function ϕ : Z+ → C, the components of θF (ϕ(N)) (whose domainscontain D0) are

θ00(ϕ(N)) = |λ|2(N + 1){ϕ(N + 1)− ϕ(N)}, θ1

1(ϕ(N)) = 0

θ10(ϕ(N)) = λ(N)[ϕ(N),W ] = θ0

1(ϕ(N))∗,

In particular, if we take ϕ(N) = C = N2 + 1, then

θ00(C) = (2N + 1)|λ|2(N + 1),

and

[C,L] = θ10(C) = λ(N)[N2,W ] = λ(N)(WW ∗ + P0)N2W − λ(N)WN2

= λ(N)W (2N + 1)

where P0 = 1−WW ∗ is the projection onto Ce0, so that P0N2W = 0. Thus [C,L] is

relatively bounded by C3/4. These observations allow us to apply Propositions 3.1and 3.4, and Theorem 3.5 to prove the existence of an isometric process U that isa strong solution to

dUt =[(− 1

2 |λ|2(N + 1)− iµ(N)) dt−W ∗λ(N) dAt + λ(N)W dA†t

]Ut.

If, further, we assume the existence of 0 < c′ ≤ c such that

c′n ≤ |λ(n)|2 ≤ c(n + 1) ∀n ≥ 0,

then Dom K = Dom N and we can apply Corollary 3.2 with M = c(N + 1) andk = c to deduce that U is a unitary process. As a particular example we takeλ(n) = −in1/2, µ(n) = 0. Then equation (R)′ reads

dUt =[− 1

2BB† dt− iB dAt − iB† dA†t

]Ut,

where B† = N1/2W and B = W ∗N1/2 are the usual creation and annihilation op-erators on h. As shown in [Wal], this equation arises by considering the interactionof an inverse oscillator in a heat bath and taking the singular coupling limit. Astrong solution of the QSDE is constructed in [Wal] by use of Maassen kernels,and analytical difficulties such as investigating the range of the Ut are surmount-able there because of the simple algebraic structure of the equation for this specialchoice of λ and µ.

Perturbations of Hamiltonian evolutions. Let h = L2(R). We shall use thefollowing vector spaces of functions on R: Ck

b (R), the space of bounded continuousfunctions with bounded continuous derivatives up to the order k; C∞

c (R), the spaceof infinitely differentiable functions with compact support; and Hk(R), the spaceof functions in h that have weak derivatives up to order k in h. Let σ, ρ and ξ (resp.η) be R-valued functions in C4

b (R) (resp. C5b (R)), m a positive constant, and define

the triple (L,H, S) as operators on H2(R) by:

Lu = σu′ + ρu, Hu = − 12m

u′′ − i

2(ηu′ + (ηu)′) + ξu, S = 1,

for u ∈ H2(R). So then L∗, restricted to H2(R), satisfies

L∗u = −σu′ + (ρ− σ′)u.

Under the assumption ‖σ2‖∞ < m−1 it can be shown that − 12L∗L + iH, re-

stricted to H4(R), is closable and that its closure is the generator of a stronglycontinuous contraction semigroup ([Kat], Theorem 2.7, p.499). Moreover its do-main coincides with H2(R). So we can apply Proposition 3.1 with D = H4(R) toobtain a contraction process U∗ that is a solution to (L)′ on D.


Define the reference operator to be the positive self-adjoint operator C withdomain H4(R) given by

Cu = u(4) + u.

Lemma 4.1. The hypotheses of Proposition 3.4 hold for the triple (L,H, S) andthe reference operator C when we put D = H8(R).

Proof. Since D = Dom C2 it follows that it is invariant under each Rε, and clearlyD is contained in the domains of L∗L,L∗, [C,L] and [C,H], so that (i) holds. Also,(iv) holds trivially.

As a first step for verifying (ii) we compute the commutator i[H,C]. Denotingby ∂ the differentiation operator,

[η∂ + ∂η,C] = [η, ∂4]∂ + ∂[η, ∂4]

= −η′∂4 − 2∂η′∂3 − 2∂2η′∂2 − 2∂3η′∂ − ∂4η′

= −3∂η′∂3 − 2∂2η′∂2 − 3∂3η′∂ + η′′∂3 − ∂3η′′

= −8∂2η′∂2 + 3∂η′′∂2 − 3∂2η′′∂ − ∂2η(3) − ∂η(3)∂ − η(3)∂2

= −8∂2η′∂2 − 4∂η(3)∂ − ∂2η(3) − η(3)∂2

and

[ξ, C] = −2∂ξ′∂2 − 2∂2ξ′∂ − ∂ξ(3) − ξ(3)∂

By the Cauchy-Schwarz inequality

|〈u, [η∂ + ∂η,C]u〉| ≤ 8‖η′‖∞‖∂2u‖2 + 4‖η(3)‖∞‖∂u‖2 + 2‖η(3)‖∞‖u‖‖∂2u‖

≤ 8‖η′‖∞‖∂2u‖2 + 6‖η(3)‖∞‖∂2u‖‖u‖

≤(8‖η′‖∞ + 3‖η(3)‖∞

)〈u, Cu〉,

and

2|〈u, [ξ, C]u〉| ≤(6‖ξ′‖∞ + 3‖ξ(3)‖∞

)〈u, Cu〉

for all u ∈ D. Thus we obtain

2|〈u, i[H,C]u〉| ≤(8‖η′‖∞ + 6‖ξ′‖∞ + 3‖η(3)‖∞ + 3‖ξ(3)‖∞

)〈u, Cu〉.

Similarly[C,L] =

∑0≤α,β≤2

∂αϕαβ∂β ,

where

ϕ22 = 4σ′, ϕ11 = σ(3), ϕ10 = ϕ01 = ρ(3), ϕ00 = 0,

ϕ12 = 2ρ′ − 2σ′′, ϕ21 = 2ρ′, ϕ20 = 0, ϕ02 = σ(3),

so again even though C is a differential operator of order 4, and L a differentialoperator of order 1, their commutator has order 4. Taking adjoints, we have

[L∗, C] =∑

0≤α,β≤2

(−1)α+β∂βϕαβ∂α.

Therefore the term of order 5 in ∂ of the differential operator

θ00(C) = [L∗, C]L + L∗[C,L]

must come from−∂σ∂2ϕ22∂

2 + ∂2ϕ22∂2σ∂,


and in fact its coefficient vanishes, so that θ00(C) is actually of order at most 4. The

same arguments used in the estimate of |〈u, i[H,C]u〉| yield the inequalities

|〈u, {[L∗, C]L + L∗[C,L]}u〉| ≤ c〈u, Cu〉, |〈v, θ10(C)u〉| ≤ c〈u, ι(C)u〉

for all u, v ∈ D (and where we have set u = [u, v]>), and for some constant c thatdepends only on σ, ρ and their derivatives up to the order 4. Thus the inequalityin part (ii) of Proposition 3.4 holds for

b3 = 3c + 4‖η′‖∞ + 3‖ξ′‖∞ + 32‖η

(3)‖∞ + 32‖ξ

(3)‖∞.

Finally, to show that (iii) holds, note that ∂α′C−1∂α is a contraction for all

0 ≤ α, α′ ≤ 2, and so

‖C−1/2[C,L]u‖2 =∑

0≤α,β,α′,β′≤2

⟨∂α′

ϕα′β′∂β′u, C−1∂αϕαβ∂βu

⟩=

∑0≤α,β,α′,β′≤2

(−1)α′⟨ϕα′β′∂β′

u, (∂α′C−1∂α)ϕαβ∂βu

⟩≤ 9c′

∑0≤β,β′≤2

‖∂β′u‖‖∂βu‖

where c′ = max0≤α,β≤2 ‖ϕαβ‖2∞. Thus

‖C−1/2[C,L]u‖2 ≤ 9c′( ∑

0≤β≤2

‖∂βu‖)2

≤ 27c′∑

0≤β≤2

‖∂βu‖2 ≤ 812 c′〈u, Cu〉.

This proves the lemma. �

Now note that Dom C1/2 ⊂ Dom− 12L∗L± iH∩Dom L∩Dom L∗, and so setting

Dε = H4(R) for all ε we see that the conditions of Theorem 3.5 hold, noting part(b) of the remarks after Theorem 2.3. Thus there is an isometric process U that isa strong solution to

dUt = (L∗ dAt − LdA†t + K dt)Ut

on Dom C1/2 = H2(R). That U is a coisometry process can be shown by applyingthe arguments of [ChF], Section 5.1, where it is shown that the QDS associated tothe model of heavy ion collision from [AlF] is conservative, this time taking M tobe a multiple of −∂2 + 1.

The above argument can be modified by taking

Hu = − i

2(ηu′ + (ηu)′),

and in this case we no longer need to impose the bound on ‖σ2‖ to show that K,the closure of − 1

2L∗L + iH, is the generator of a strongly continuous contractionsemigroup (see, for example, [F5], Theorem A.3). The rest of the calculation abovecan then be applied directly to prove the existence of an isometric solution to (R)′

for this new form of H. Moreover it can be shown as in [F5], Chapter 4, thatthis solution is unitary. If we consider the algebra L∞(R) acting by pointwisemultiplication on h, then since we have

θ00(f) =

12σ2f ′′ + (σσ′ − σρ + η)f ′,

we see that the flow X 7→ U∗t (X ⊗ 1)Ut gives a realisation of a diffusion process

with covariance σ and drift σσ′ − σρ + η.

Acknowledgement. The majority of this work was carried out during a visit bySJW to the Universita di Genova, supported by a Junior Research Fellowship from


the CNR, and completed while at the Universite de Paris VI, supported by an EUgrant from the TMR Network for Noncommutative Geometry.

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(FF) Dipartimento di Matematica, Universita di Genova, Via Dodecaneso 35, I-16146

GenovaE-mail address: [email protected]

(SJW) Department of Mathematics, University College, Cork, IrelandE-mail address: [email protected]

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