tional assumptions one can formulate necessary and sufficient conditions which are analogous to Feller's condition for regularity of a diffusion process: the ...
21. 22. 23. 24. 25. 26.
S. Johansen, "An application of extreme point method to the representation of infinitely divisible distributions," Z. Wahrscheinlichkeitst. Geb., 5, 304-316 (1966). G. Lindblad, "On the generators of quantum dynamical semigroups," Commun. Math. Phys., 48, No. 2, 119-130 (1976). B. Misra and E. C. G. Sudarshan, "The Zeno's paradox in quantum theory," J. Math. Phys., 18, No. 4, 756-763 (1977). M~. Ozawa, "Quantum measuring processes of continuous observables," J. Math. Phys., 25, No. I, 79-87 (1984). K. R. Parthasarathy and K. Schmidt, Positive Definite Kernels, Continuous Tensor Products, and Central Limit Theorems, Springer, Berlin (1972). K. R. Parthasarathy, "One parameter semigropus of completely positive maps on groups arising from quantum stochastic differential equations,"~!Bull. Unione Matem. Italiana,
5_A, 57-66 (1986). 27. 28. 29. 30.
31.
Quantum Probability and Applications II, Springer, Berlin (1985). D. Russo and H. A. Dye, "A note on unitary operators in C*-algebras," Duke Math. J., 33, 413-416 (1966). I. J. Schoenberg, "Metric spaces and positive definite functions," Trans. Am. Math. Soc., 4, 522-530 (1938). M. Sch~rmann, "Positive and conditionally positive linear functionals," in: Quantum Probability and Applications II, Springer, Berlin (1985), pp. 474-492. pp. 474-492. M. Takesaki, Theory of Operator Algebras, Springer, New York (1979).
NECESSARY AND SUFFICIENT CONDITIONS FOR CONSERVATIVENESS OF DYNAMICAL SEMIGROUPS UDC 517~986.7
A. M. Chebotarev
Dynamical semigroups constitute a quantum-mechanical generalization of Markov semigroups, a concept familiar from the theory of stochastic processes. Let ~ be a Hilbert space and ~ a yon Neumann algebra. A dynamical semigroup Pt is a oweakly continuous one-parameter semigroup of completely positive maps of ~ into itself. A semigroup Pt possessing the property of preserving the identity f E ~ is said to be conservative and its infinitesimal operator L['] is said to be regular. The present paper studies necessary and sufficient conditions for strongly continuous dynamical semigroups to be conservative. It is shown that under certain additional assumptions one can formulate necessary and sufficient conditions which are analogous to Feller's condition for regularity of a diffusion process: the equation P = L[P] has no solutions in ~ + . Using a Jensen-type inequality for completely positive maps, constructive sufficient conditions are obtained for conservativeness, in the form of inequalities for commutators. The restriction of a dynamical subgroup to an Abelian subalgebra of ~=(R ~) yields a series of new regularity conditions for both diffusion and jump processes.
INTRODUCTION The notion of a dynamical semigroup generalizes that of the semigroup generated by a Markov process. It is used to describe the evolution of observables in open quantum systems, as well as processes of interaction between micro- and macroscopic objects. The mathematical definition of a dynamical semigroup may be found in Lindblad [21] and in Gorini et al. [15]. Let ~ be a Hilbert space and ~ { ~ ) the Banach space of nuclear operators acting in I~. A dynamical semigroup on ~ ( ~ } is a strongly continuous one-parameter!contraction semigroup of linear completely positive trace-preserving maps~ Yt : ~ - { ~ ) - ~ { ~ ) The family of adjoint maps Pt of a v o n Neumann a l g e b r a ~ into itself determines a o-weakly continuous adjoint Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 36, pp. 149-184, 1990.
0090-4104/91/5605-2697512.50
9 1991 Plenum Publishing Corporation
2697
dynamical semigroup of normal completely positive contraction maps which preserve the identity of ~ [15, 21]. The semigroup T t describes the evolution of the states and corresponds to the SchrSdinger picture of quantum mechanics, whereas Pt describes the evolution of the observables in the Heisenberg picture. The semigroups T t and Pt possesses the respective trace- and identity-preserving properties either simultaneously or not at all [15]. For semigroups with unbounded infinitesimal operators this property is nontrivial, and some authors do not include it in the definition of a dynamical semigroup [2]. Semigroups Pt and T t with the identity- and trace-preserving property, respectively, are known as conservative semigroups and their infinitesimal operators are said to be regular. Davies [16] considered an example of a strongly continuous conservative dynamical semigroup T t related to the model of quantum neutron diffusion accompanied by neutron scattering and absorption, with two neutrons cerated for each one absorbed. His result is an operator generalization of the regularity theorem for two-sided Poisson processes in which the intensity of the jumps increases at a linear rate. In physical terms, the failure of a system with a variable number of particles to the conservative means that the number of descendants of an individual particle blows up to infinity in a finite time and with nonzero probability. This phenomenon is interpreted as an explosion. Using the construction of a minimal dynamical semigroup, Davies derived the estimate Tr Tt(p)~Tr p for a contraction semigroup T t and showed that in the neutron diffusion problem one also has the converse inequality, which together with the previous inequality guarantees that T t is conservative. The semigroup Pt is adjoint to T t with respect to the bilinear form = Tr pX, = , which separates the elements of the cone ~+. Therefore TrTt(p) = T r ~ if Pt(1) = 1 and, conversely, Pt(1) = I if TrTt(p) = Trp. In this paper it will be shown that a necessary and sufficient condition for a strongly continuous dynamical semigroup Pt to be conservative is that .~+ contain no solutions of the equation X = ~ ( X ) , where ~(.) is the infinitesimal operator of the semigroup in the Heisenberg representation and ~ + the cone of positive elements of ~. This conservativeness criterion is related to Feller's condition for the regularity of diffusion processes (see [5]) and is considerably simpler in form than other conditions presented in Sec. 3, but it is by no means always directly verifiable. Constructive sufficient conditions for conservativenes will be presented in Sec. 4. For semigroups generated by stochastic processes, conservativeness conditions are also of some interest, since the property of being conservative excludes such phenomena as evolution of the process to infinity or blowup of the number of jumps to infinity in a finite time with nonzero probability. Necessary and sufficient conditions for the conservativeness of semigroups generated by a one-dimensional diffusion process and a one-way Poisson process were derived by Feller (see [5]). A corresponding condition for regularity of Markov jump processes in which the jump intensity is almost everywhere finite but unbounded from above on the phase space was derived by Gikhman and Skorokhod [3]. Constructive sufficient conditions for regularity of nondegenerate diffusion processes may be found in the work of Khas'minskii [9] and Ichihara [17]. The regularity of degenerate diffusion processes with hypoelliptic generating operator has been studied by Malliavin [22], Ikeda and Watanabe [18], Stroock [21] and Bismut [12]. Later, Bismut used his method to derive sufficient conditions for the regularity of degenerate jump processes with independent increments whose intensity is unbounded on sets of jumps approaching zero [13]. Generalizations of his approach to inhomogeneous Markov processes were considered by L~andre [19]. Skorokhod and the present author have obtained sufficient conditions for regularity of inhomogeneous jump Markov processes with jump intensity unbounded from above on the phase space of the stochastic process. As shown in [ii], the inequalities used in the Abelian case generalize to operator algebras and can be used to obtain sufficient conditions for strongly continuous dynamical semigroups with unbounded infinitesimal operators to be conservative. In this paper we shall derive conditions for strongly continuous dynamical semigroups to be conservative. The infinitesimal o p e r a t o r ~ ( - ) bounded normal map of the algebra ~ is as follows:
2698
of a norm continuous dynamical semigroup Pt (') is a into itself. As shown by Lindblad [21], its structure
~(X)=4)(X)--~.Xq-~H,X],
where ~(') is a bounded linear completely positive map of .~ into ~, tity operator, [A, B] = AB - BA is the commutator, A.B = (AB - B A ) / 2 of operators. The operators ~ and H appearing in the expression for tor ~(.) of a norm continuous semigroup Pt(') are members of ~ and s e m i g r o u p W , = e x p { - - G ~ , G-.~-.iH.+-t/I(I), acting i n ~ .
O=4)(I), I is the idenis the Jordan product the infinitesimal operagenerate a contraction
The evolution equation for Pt(') (Lindblad's equation)
ff~Pt(X)-.~? (P,(X~, P,(X) [,..c=.X may be written in an equivalent integral form
P, (x)= w;xw, + I asW;_;| (p,
w,_,.
( o. )
0
where W~ is the adjoint semigroup, with generator G'-~---iHq-Z/~O. The integral form of Lindblad's equation involves explicitly only part of the map ~L~(-), a circumstance which simplifies the interpretation of the right-hand side of the equation and its domain of definition if the map ~(.) and operator H are unbounded. The difficulties arising in the theory of dynamical semigroups with unbounded infinitesimal operator stem, first and foremost, from the fact that the domain of the maps ~(.) and ~(.) may not be an algebra [I, 2, 14]. This difficulty is avoided in [Ii] and in this paper, first, by replacing the operators ~(.) on the right of Eq. (0.i) with the associated bilinear forms, which are defined in a larger domain, and, second, by "internal" completion of the set of maps of the algebra ~ to a set of bounded bilinear forms. The completion is accomplished using a topology which guarantees inclusion of the algebra ~ in the domain of all the bilinear forms - unbounded ones included - that belong to the completion. Let us explain how the right-hand side of Eq. (0.i) is defined in terms of bilinear forms. To each bounded operator X6~(8~) one can associate, in a natural manner, a bilinear form X, o n ~ X ~ : X . ~ , ~ ] = ( ~ , X ~ ) , and conversely, to each bounded bilinear form Y, one can associate an operator [Y.]6~(~) such that [Y,] = Y [4, p. 256]. Henceforth the subscript * will be used to designate the operation associating a bilinear form to an operator, whereas the brackets [-] stand for the inverse operation. Thus, if the infinitesimal operator is bounded, the integral term in Eq. (0.I) can be written in one of two equivalent forms:
(x))
dsWL r (P,
w,.,
The natural correspondence between bilinear forms and linear operators in Hilbert space extends to the set of closable bilinear forms which are bounded from below and the set of self-adjoint operators which are bounded from below. The domain of the associated bilinear form is larger: ~9(~.)~-~9([O.]'t~)~9([~.]) (see [4, p. 323]), and therefore, throughout the sequel, we shall describe dynamical semigroups in terms of Eq. (0.1) in which the right-hand side involves an integral of bilinear forms: r
The c o n d i t i o n s imposed b e l o w on t h e i n f i n i t e s i m a l o p e r a t o r g u a r a n t e e t h e e x i s t e n c e of a m i n i m a l r e s o l v e n t o p e r a t o r o f Eq. ( 0 . 2 ) w h i c h i s s t r o n g l y c o n t i n u o u s and n o r m a l . The main p r o p o s i t i o n s and r e s u l t s w i l l now be s t a t e d .
i.
FORMULATION OF MAIN RESULTS
In this paper we shall use the notation of [ii]: ~ is a Hilbert space with scalar product (.,.) which is antilinear in the first argument, ~ a certain yon N e u m a ~ algebra of bounded linear operators acting in ~, d~g+ the cone of nonnegative Hermitian operators generating ~ and das the set of operators in ~ § majorized by the identity operator 16~. The domain of the bilinear form X, associated with an unbounded operator !X : ~ ( X ) - ~ will be denoted by ~9{X,), and the value of the corresponding quadratic form of an element ~E~)(X.) by X,[~]. To each positive densely defined symmetric operator X : ~ ) ( X ) - ~ one can associate a positive densely defined closable form X., ~)(X.)~Z)(X) [4, p. 318], and to each positive closable densely defined bi!inear form ~ . : ~ 9 ( ~ ' ) X ~ { ~ . ) - ~ C a positive self-adjoint operator, denoted by [~.]:~9(O.) =~[O.]11=~)[~.] [4, p. 323]. To an element Q from the algebra 2699
9.~(.~, ~[~')of bounded linear maps of ~ set of bounded bilinear forms:
into ~
we associate a linear map
Q.(A)[(p, @]-----((gQ(A)@),
Q.
of ~
into the
A(~.
The set of all such maps is denoted by ~'.(~,~). Maps in ~'. need be defined only on the generating cone of nonnegative Hermitian operators ~+; whose associated nonnegative bilinear forms constitute the set ~ . + ( ~ ) . The indexes w and s will be used in the rest of this exposition to denote weak and strong convergence; ~-"(~, ~ ) will denote the linear space of (w, s)-continuous maps of ~ into ~ :
P(Xn)-+O , if
Vk'n~'~0, P~s('9~,~),
{~n}~,
S
and --, , , ~ ) the corresponding linear space of maps of ~
into the set of bilinear forms.
Let I~"I~ be the norm in ~. generated by the scalar product and ~ a Banach space, dense in ~ , with norm II'il~ll'U 9 Let us consider the completion of the set ~ ' ~ ( ~ , ~ ) in the separable locally convex topology determined by the system of seminorms PA,B(" ) of parametrized absolutely convex (i.e., symmetric) sets ASz~ and B6~: which are compact in the strong topology on ~ and the uniform convergence topology on ~ , respectively:
The completion of the linear space ~r~"(~, ~ ) in this topology will be denoted by ~ " ( ~ , ,~). An analogous construction of the completion of the space ~ - , { ~ , ~ ) was used in [Ii]. A map @, in ~m~,,(~, ~
is said to be completely positive (CP) if
for any finite tuples {Yj}Es~, {~j}6~i Using Stinespring's tion just described, one under composition to the which is due to Kholevo,
representation [23] of bounded CP maps and the completion construccan show that the set of ws-continuous CP maps CP.~(~,~) is closed right with bounded maps in CP.~'(.9~,~). An outline of the proof, may be found in [ii].
In the sequel it will be assumed tire bilinear form and @ = @(I) is the main dense in ~Z)(~)~(~).). We shall is not a mathematical curiosity. For infinitesimal operators
that ~.(.)6CP."~(~I,~), ~.=~.(I) is a closable nonnegaassociated nonnegative self-adjoint operator with donot consider the case ~ D ( ~ ) = ~ , which, incidentally, example, the Markov semigroups studied in [13, 19] have
N
(x) =
I a g, (z) {e'"Vxe-'",'- x -
[He,xD + [Ho,Xl,
where {Hk} N are first-order self-adjoint operators, gk:R ~ R t, ~g~ ~z).mln (I, z D dz < ~. --oo
The theory presented below will be extended to such equations in Sec. 5. As Lindblad's equation is invariant under the substitution @ + @ + I.~, we may assume without loss of generality that ~ (I)91 and demand that the domain of the self-adjoint operator H appearing in ~{') meet ~Z)(~) in a dense subset o f ~ :
where
"~o
is the closure in ~
of ~0"
We shall assume moreover that there exists in ~ a (strongly) continuous contraction semigroup of operators W t with generator G, such that
2700
where ~
is the Banach space used in the completion procedure applied to the s e t ~ r ~ s ( ~ , ~ -
To ensure that for every t the operator W t be an element of the algebra i~r we require that for all X > 0 ~ r should contain the resolvent of G:
(O+XI)-~,~.
(1.3c)
A sufficient condition for the choice of the Banach space ~ to be compatible with the strong continuity property of the solutions of Lindblad's equation is that W t be bounded and strongly continuous in ~:
ttw , , lt~ I, C is some positive constant, and [H, ~-z], and [r on ~ ( O ) given by the rule:
[H, O-'l.(~, ,) = ( ~ ,
(1.7)
H], are bilinear forms
o - ~ , ) - - ( o 7 % n , ) : [~, n ] . ( ~ , ) ~ = ( ~ , H,),CH~; O,).
Conditions (1.7) w i l l hold i f r i s a bounded map and H a bounded o p e r a t o r . If r ~(G)-~(H), t h e second c o n d i t i o n i s a c o r o l l a r y of t h e f i r s t . The f i r s t i n e q u a l i t y ( 1 . 7 ) may also be phrased as an inequality for bilinear forms:
o, (o,0" (o~), < co,, o,, ~ r (0 (I 4- xo CO)-', where (~2), is a bilinear form with domain ~(~)), defined by the rule: (iD=).[~p,~]=((D~,~-).
2701
2.
INTEGRAL FORM OF LINDBLAD'S EQUATION
In this section a rigorous meaning will be given to the right-hand side of Eq. (0.2). As we shall show presently, for every X 6 ~ the integral on the right of (0.2) is a bounded family of operators, strongly continuous as t + 0. The strong continuity of this family follows from its weak continuity and boundedness; its weak continuity, in turn, is a consequence of the choice of the system of seminorms (i.i). Let ~),~CP;~(~, ~),
X.~-~aXt in
~,
~-~
in S
as ~-~t%rtE~+, and let U be a compact neigh-
borhood of t. The closed absolutely convex envelopes of the compact sets {X~},Eu, {~),@u,, denoted by A and B, are compact in ~ and ~ , respectively [7, p. 60]. It follows from the definition of the topological vector space ~r~(~, ~) that for any ~,6CPy;~.~, ~i and any compact sets A ~ , B ~ @ there exist a sequence {@,)6CP~(~, ~ ) c o n v e r g i n g to ~, in 8v~s(~, ~) and a sequence en = gn(A, B), en > 0 converging to zero such that
sup
I (~
(x)- ~,(x))[~, ~11 ~ ~.
Since as 9 § t we have l[m~)jv~X~)I~, ~]==~)N(Xt)[~,~p], I|m~N{X)[~,~,]=~N(X)[~, ~,], it follows that
lira sup [ @, ~(X,-- X,) [~, ~l I = O, if
,.,,.,~
limsupt|
if
X.~-..,,X, in ~,9
(2. i)
#~-+~, i n S .
~, *6s It follows from (2.1)" that for a strongly continuous family of operator in ~ , say {X,}, s@R+, X~6~, X~ : ~ - + ~ and a strongly continuous family of operators in ~ , say {W~}, s~R+, W~6~, ~ , : S - ~ , the bilinear form ~.(X~)[Wt-~8, W,_~], 8, ~69 is continuous in s%ID,~] and t6~+. Hence there exists a symmetric bilinear form ~t on ~ X ~ equal to the integral of the bilinear forms : f
,~,(x) I,,~l=; ,~s~.(p, (x))[w,_,~,~,_,,~I, 0
where Ps(") is any strongly continuous bounded family of linear maps of ~%~ into ~ Since W , : ~ ( G ) - ~ ) ( 6 )
and
Olm(o)-~Hq--I @,
and X ~ .
where H and ~ = ~(I) are self-adjoint opera-
tors, it follows that for any ~ ) ( G )
T h e r e f o r e , t h e v a l u e s of t h e q u a d r a t i c form 9~(X)[.] on t h e s e t ~ 9 ( G ) ~ S , which is dense in ~ , s a t i s f y an e s t i m a t e which shows t h a t ~p~[.] i s bounded in gg:
1~,(x) I,ll 0 the map
~)~(.)=e-~ -~,'~, ~)=~(I)>I ~, is an element of CPw*(ag, ;ld), then ~.(.)~CP,'~'~(at,.~)~ ,~':' 2703
Proof. We shall show that ~n(') is a Cauchy sequence in the topology defined by the system of seminorms (i.i). Let A and B be compact sets in ~ + and ~ , respectively. Then, f o r any X6A, ~6B ,
Iiq~, (~'. i x ) - 6,. i x ) ) , I~:11e-r
(X) q~I[ II r
(X) (e-'~t= - e-*t,.), II~
• It. , n (x) e - ~ - , II< IIX II {ll ~ II~"II(e-|
e-|
~,n (X) (e-| " - e.|
IIJ + II* IIj" II(e-~
-- e-|
q, IIX
e liar}.
Therefore, lira
sup
where ~r =-~l"z~6a~,
i(qJ, (:9#,(X)-~n(X))qil'~:>0,
8 "
~i' o r
H =---2~-m A + eV (r},
o +r, vea=,s~(gt),
where .~(o1~) is the algebra of all bounded linear operators acting in g~. It was shown in [ii] that conditions (1.7) will hold if c = 4 ~ ( l + 2 / x ) , for any ~ > 0. We claim that @,(.)6CP:'~(~,~), where ~ is the Banach space of functions of finite norm II~[I~---II~112~II. The choice of the Banach space ~ together with the third condition (1.7) guarantees validity of the compatibility condition (1.4) (see Lemma 2.1) and condition (l.3b). Condition (l.3a) is also fulfilled: C~(/~==~oC.~(H)N.~(@), ~ o = a ~ . The half-plane C_-----{z: Rez 0 [condition (1.3)]. show that ~, (.)@CP~,~ X(ar ~).
We use Lemma 2.2 to
8
In this example,
(~,~(X,)~)=X(~j,g , Xng7,~), where ~Eaw~, ~/,~==Ox~s
and
II*;,~I!< II*IIsup (pe-~'~")=II~ II.o(Vf): v6a§ If the sequence {Xn} is weakly convergent to zero, then ~(~/.~, X'n~i,x)-~0. (~r ~). Using Lemma 2.2, we conclude that @,(.)6CP:~(~, 5~). 3.
Therefore.~(.~CP "~ •
NECESSARY AND SUFFICIENT CONDITION FOR CONSERVATIVENESS OF MINIMAL SOLUTION
Consider the following recurrent sequence of bounded CP linear maps
p~~
p~~ ( x ) - w;xw,
(3.1)
where O,0j~P~z(.#, ~). I t was proved in [11] t h a t if~X~.~ § then the sequence of o p e r a t o r s P~n)(x) i s monotone in two senses. F i r s t , i t i n c r e a s e s m o n o t o n i c a l l y with i n c r e a s i n g n: O.Q (/) ~0, Q- (O-Q'+' (I) =Q" ( I - Q (0)/>0, so that the sequence {Q(1)} l belongs t o ~ z and is monotone decreasing. By the Least Upper Bound Lemma, there exists a trivial or nontrivial limit X=s--lim Q~(I). If X ~ 0, then X is a nontrivial solution of both equations: X = Q ( X ) . X.=~.(X). Conversely, if the equation Y = Q(Y) has a nontrivial solution in .~§ then X = Y/IIYII is a solution in ~t of the same equation and therefore O~Q"(I--X)=Q"(1)--X. Since Q~(1)/>X for every n, it follows by the Least Upper Bound Lemma that there exists a nontrivial limit y = s - l i m Q-(1). This proves the theorem. 2708
The necessary and sufficient conditions for conservativeness as formulated in Theorem 3.3 are formally simple, but from the point of view of applications the test established in Theorem 3.2 is more convenient. We demonstrate with an example~ Let ~=~(f), ~ > ~ ) ,
and let W t be the strongly continuous semigroup generated by the
operator ~=iH~_l.{~q-~(~i--~)),
%~[0> I]~ where ~)((7~)~(~i), ~)(~i)=-~.
The relations
dt
= d-i
= e-'
dt
[W,x~,
=
show that
re-'
--
0
.
,
0
AI~==6[I~-!(IIF~)ilJF~e#is
monotone decreasing, while
(w,) (|174
e
=s-~
is a monotone increasing function of the parameter ~[0, I]:
Using this inequality, we consider an example of a strongly continuous conservative dynamical semigroup, due to Davies [16]. o o
Example 3.1. operators: ~--I, W t, and let
Let ~
be a Hilbert space and {~k}0 an increasing family of projection Assume that the subspaces ~ are invariant under the action of
~lf~l--~l~iflt'.
+
_
_
=~i (X)=,-=,| (=~+,X=~+,)=~ V X ~ where {lk} is an increasing sequence of positive numbers. Then, on the one hand, for any ~ h
(%. Q-+' ( l ) ~ ) - ( , . On the other hand, the inequality imply the estimate:
:
~ Q (ah+,Q ( .. ; sh+.Q (/)~h+,~ 9 9 .) ~ + ~ ) a ~ ) .
n~.l(/)nh+.~kh+.l
and inequality (3.4) f o r ~ l . : ~ i ~ - - % ~
< ? e-,,(,--
,'-.-,
Therefore,
The s e t ~ i ~ U ~ t ~
i s dens e i n ~ ,
and so s-ltmQ~(1)=O i f t h e s e r i e s
11
Thus,.~.i"l~ ~
x,-L. ~ . i "l i s d i v e r g e n t .
n
is a sufficient
The a s s u m p t i o n s l i s t e d
condition for conservativeness.
above c o n c e r n i n g t h e p r o p e r t i e s
group W t hold for the Fock space ~ = $ ~ N o
of t h e map 0 ( ' )
~#~,v~C~)
and ~the semi-
and the map ~ { ' ) i ' ~
0
~)J(') [16]:
I-4
(/,)=(/:), S s(.)6~ (0%
/ j (q) = is (x + e),
where @(R") is the space of smooth rapidly decreasing functions and H = -A. The c o r r e s p o n d i n g d y n a m i c a l semigroup d e s c r i b e s a model o f t h e e v o l u t i o n of o b s e r v a b l e s in an open quantum system, where the operator B x characterizes absorption of neutrons, the 2709
2
3
the operator B x scattering and the operator B x the creation of two neutrons upon the absorption of one. The estimate is based on a lemma of Hoegh-Krohn, cited in [16]. It follows from this estimate that the sufficient condition for conservativeness is satisfied:~-t~(C~)-|=~.
x~)~O , therefore
{~,A~)=~e-t[IV/t~;N2dt>O,i.e.,
kerA = {0}.
We
s;0
claim that A(~9(O~) is dense in ~ . Indeed, suppose the contrary: the linear space L = A-(~Z))(O)) is not dense in ~ , i.e., there exist h ~ and an open convex neighborhood U = U(h) of h disjoint from L. Then it follows from the H a h n - B a n a c h theorem that there is an element ~6~, ~ 0 , which is orthogonal to L and positive on U. Let {~n}~ be a sequence of elements in /~)(G) converging in ~ to ~. Then, on the one hand, |im(~.A~,) = ( ~ , A ~ ) > 0 , because ~ ~ 0 and the operator A is bounded and positive: on the other, A~n~L, and therefore (~, A~ n) = 0 for all n. This contradiction implies that L'=~. Thus, A -l is a densely defined symmetric operator: its domain ~(A-') contains a dense subset A(~9(G)) of ~ , while its range contains all of ~. Consequently, A -i is a self-adjoint operator [4, p. 269], A-l~>2l. The positive operator is also self-adjoint. Simple calculations produce the last statement of the lemma: Q(I) = I - A = A.B = (B + I)-iB = (I + B-i) -I This completes the proof. In the sequel this result will be used to estimate the powers of the map Qn(I).
B---A-I~I>~I
4.
INEQUALITIES FOR COMPLETELY POSITIVE MAPS
Upper and lower estimates for sequences of positive operators {Qn(I)}~ follow from a well-known inequality for CP maps:
Q(z)>/Q(1) (Q(z-') )-'Q(1),
z~..~+,
(4.1)
which is valid if the operators z and Q(z -~) are invertible and z -~ is bounded (see [20]). In particular, inequality (4.1) implies an inequality which will be used below to derive necessary conditions for conservativeness: Q ((!+ F)-')~ Q (f) ((Q (I)+ Q (Y))-'~(f) -- (~(f) q- Q(F))-', where (Y)= Q-' (I) Q (Y) Q " 41).
(4.2)
Let Q(') be the map defined by (3.3). Then it follows from Lemma 3.3 that Q(I) = (I + B-~) -~, where B is a positive self-adjoint operator with If the operatorY is chosen in such a way that Y = B -~, then Q(I) = I + B -i = I + Y. Thus
B'~l, ,/uI~Q(1)
-~
O,
(4.14)
-I. A dynamical semithen Qn(B~.) 0 for which
D,(p)+ D~ (p),;cA (p), DO(p)+ Dt (p) A-Z(p)Dt ~).~ c (V ~),{- l). As (p)+ A~ (p)a; cA (p),
(5. I)
At 0~)+ At (p) A'* ($) At @) ~ r (V (P}+ 1)" In p a r t i c u l a r , i f H = 0, f = 0, c = 0, then in the s i m p l e s t , one-dimensional case, conditions (5.1) reduce to the i n e q u a l i t y supa(p)if'(p)--cq~ (I). We claim that this condition will be satisfied if de f
o
