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Nondissipative curves in Hilbert spaces having a limit of the corresponding correlation function. (English) Zbl 1001.47022

The authors consider a class of nondissipative basic operators, denoted by \(\widetilde\Omega_R\), that are a coupling of a dissipative operator and an antidissipative one. Associated with the family \(\widetilde\Omega_R\), a class of nondissipative curves in Hilbert spaces, whose correlation functions have limits at \(\pm\infty\), is presented. The work under review is a continuation and a generalization of some investigations due to K. Kirichev and V. Zolotarev [see Integral Equations Oper. Theory 19, No. 4, 447-457 (1994; Zbl 0886.60027) and ibid. 19, No. 3, 270-289 (1994; Zbl 0805.60031)] on the model representations of curves in Hilbert spaces where the corresponding semigroup generator is a dissipative operator. Among other results, the wave operators and the scattering operator for the couple \((A,A^*),A\in\widetilde\Omega_R\), are obtained, The authors’ results show an interesting phenomenon: the nondissipative case under discussion is near to a dissipative one.

MSC:

47B44 Linear accretive operators, dissipative operators, etc.
47A40 Scattering theory of linear operators
47A48 Operator colligations (= nodes), vessels, linear systems, characteristic functions, realizations, etc.
60G12 General second-order stochastic processes
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