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Multiplicative perturbations of \(C\)-regularized resolvent families. (English) Zbl 1114.47300

\(C\)-regularized resolvent families are generalizations of resolvent families which are solutions of integro-differential Volterra equations. The authors prove in this note a multiplicative perturbation theorem for \(C\)-regularized resolvent families generalizing the one obtained by A.Rhandi [Proc.Am.Math.Soc.119, 493–501 (1993; Zbl 0791.45005)]. Moreover, they obtain a perturbation theorem for \(C\)-regularized resolvent families. This result is a generalization of one obtained for \(C_0\)-semigroups by W.Desch and W.Schappacher [Ann.Sc.Norm.Super.Pisa, Cl.Sci., IV.Ser.11, 327-341 (1984; Zbl 0556.47022)], who introduced the space \(Z\) of admissible perturbations. One has to note that the authors use the same space \(Z\) and the same techniques without mentioning or citing this paper.

MSC:

47A55 Perturbation theory of linear operators
47A10 Spectrum, resolvent
47D60 \(C\)-semigroups, regularized semigroups
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References:

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