The existence problem for dynamics of dissipative systems in(16)

Motivated by existence problems for dissipative systems arising naturally in lattice models from quantum statistical mechanics, we consider the following $C^{\ast}$-algebraic setting: A given hermitian dissipative mapping $\delta$ is densely defined in a u

byadissipativeHilbert-spaceoperatorLintherepresentationπω.Assumemoreoverthatπωisfaithful,andthatL =0,where denotesthecyclicvectorintheGNSrepresentation.Thenδiscompletelydissipativeonitsdomain.

Proof.LetH=HωbetheHilbertspaceofthefaithfulrepresentationπωandletLbetheoperatorinHwhichisassumedtoexist,satisfyingconditions(i)and(ii)below:(i)ThedomainofLisπω(D(δ)) ,andLisadissipativeoperatorintheHilbertspaceH;(ii)Limplementsδintherepresentationπω,whichisequivalenttotherequirementthat

L isde nedonπω(D(δ)) ,andthatonthisdomainthefollowingoperatoridentityisvalid:

π(δ(a))=Lπ(a)+π(a)L foralla∈D(δ).(IX.1)

Weshow rstthatδmustnecessarilybeadissipativeoperator.Indeed,byPhillips’s

ofLexistswhichisthein nitesimalgeneratoroftheorem[30,Thm.1.1.3]anextensionL

astronglycontinuoussemigroupS(t)ofcontractionoperatorsintheHilbertspaceH.WenotethatS(t)implementsasemigroupσ(t)ofpositivemappingsinB(H),givenby

σ(t)(A)=S(t)AS(t) (IX.2)

forallt∈[0,∞)andA∈B(H).Bysemigrouptheorywenotethatthegenerator(ζsay)ofσ(t)isdissipative,sothefollowingestimateholds:

A αζ(A) ≥ A

forallα∈[0,∞)andA∈D(ζ).

Ifδωdenotestheoperatorπω(a)→πω(δ(a))withdomainπω(D(δ)),thenweclaim(easyproof)that

δω(A)=ζ(A)forallA∈D(δω),(IX.4)(IX.3)

andtheknownestimate(IX.3)abovethenimplies

πω(a) απω(δ(a)) ≥ πω(a) (IX.5)

fora∈D(δ)andα∈[0,∞).Butπωisfaithful(andhenceisometric),so(IX.5)isinfactequivalenttothedissipationestimate

a αδ(a) ≥ a