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Spectral inclusions and stability results for strongly continuous semigroups. (English) Zbl 1068.47045

In [C. R. Acad. Sci., Paris, Sér. I, Math. 333, No. 7, 641–644 (2001; Zbl 1015.47024)], the authors studied the regular spectrum for strongly continuous semigroups \(T(t)= e^{At}\) \((t\geq 0)\) and proved some spectral inclusion and stability results. As a continuation, the present paper also deals with spectral inclusions and strong stability. In the first part of the paper, it is shown that the spectral inclusion \[ e^{t\nu(A)}\subseteq \nu(T(t))\setminus\{0\} \] remains true for the regular, the left essential spectrum, and the essentially regular spectrum \(\nu\). The authors also give necessary and sufficient conditions for \(A\) to be semiregular or essentially semiregular. The second part of the paper contains some stability results; in particular, a spectral characterization of strong stability for the ultrapower extension of \(T(t)\).

MSC:

47D03 Groups and semigroups of linear operators
47A10 Spectrum, resolvent

Citations:

Zbl 1015.47024
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