The existence problem for dynamics of dissipative systems in(10)

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

VI. GENERATIONPROPERTIESOFδ

isclosedanddenselyde nedintheσ-topologyofM.But(I δ ) 1=F0,Theoperatorδ

(x) ≥ x forallx∈D(δ ).Weproceedtoshowthatinfactsowealsohave x δ

(x) ≥k x kx δ(VI.1)

).Indeed,letΛdenotethesetofk>0suchthattheinequalityforallk>0andx∈D(δ

).Thenwehaveseenthatk=1belongstoΛ.Itturnsout(VI.1)issatis edforallx∈D(δ

thatΛisbothopenandclosedasasubsetofR+,andourresultfollowsbyconnectedness.Toshowopenness,suppose rstthatk0∈Λ,andthatk∈R+satis es|k k0|<k0.We

) 1,thanuse(VI.1),fork0,inestimatingthetermsintheNeumannexpansionfor(kI δ

takenaroundthepointk0.Duetotheassumption|k k0|<k0,theNeumannseriesis

)tokI δ .Termwiseestimationconvergent,anddoesindeedde neaboundedinverseR(k,δ

) ≤k 1,anditfollowsthat(VI.1)issatis edinaneighborhoodofk0.gives R(k,δ

Considernextaconvergentsequenceofpointskn→k0withkn∈Λandk0∈R+.By

)=(knI δ ) 1exist,andtheythereforesatisfyassumptiontheresolventoperatorsR(kn,δ

theresolventidentity:

) R(km,δ )=(kn km)R(kn,δ )R(km,δ ),R(kn,δ

)∈ ) ≤k 1.Itfollowsthatthenorm-limitR =limnR(kn,δaswellastheestimate R(kn,δn

.The de nesaboundedinversetok0I δL(M)exists,anditistrivialtocheckthatR

≤k 1.HenceΛisclosed,andtheestimate(VI.1)fork0isnowimpliedinthelimitby R0

argumentiscompleted.

inMisdissipativeandclosedintheσ(M,A′)-Wehaveshownthattheoperatorδ

)topology.Itis,ofcourse,alsoclosedinthenorm-topology,anditcanbeshowrthatD(δ

isthein nitesimalgeneratorisnorm-dense.Itfollowsbysemigrouptheory[23,29]thatδ

ofastronglycontinuoussemigroupτt(0≤t<∞)ofcontractionoperatorsintheBanachspaceM.

Toshowthateachτtisanormaltransformationweconsidertheadjointsemigroupτt′(cf.[16])inthenorm-dualM′andshowthatτt′leavesA′invariant.NotethatA′isbeingidenti edwiththepredualoftheW -algebraM,sothatwemayregarditasasubspaceofM′.