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Semigroups of linear operators in a Banach space. (English) Zbl 0234.47042

MSC:
47D03 Groups and semigroups of linear operators
47A10 Spectrum, resolvent
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
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[27] , An operator-theoretical integration of the wave equation, /. Math. Soc. Japan, 8 (1956), 79-92. Notes added in proof: 1. Mr. Konishi has called the attention of the author to a paper by M. Sova, ”Probleme de Cauchy pour equations hyperboliques operationnelles a coefficients constants non-bornes”, Ann. Scuola Norm. Sup. Pisa, 22 (1968), 67-100, which contains, among others, similar results to Lemmas 2.4 and 2.7 and Theorem 4.3. 2. In Theorem 5.4 we gave two sufficient conditions for a linear operator in X to be the infinitesimal generator of an R.D.S.G. But, it can be proved that those are also necessary conditions. Hence, two kinds of characterizations of R.D.S.G. are obtained and the result gives a straightforward generalization of Theorem 5.5.
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