## 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

### Keywords:

strong stability; spectral inclusion; ultrapower extension

Zbl 1015.47024
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