Vol. 109, No. 2, 1983

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A note on fractional derivatives of semigroups and cosine functions

Hector O. Fattorini

Vol. 109 (1983), No. 2, 335–347

It was proved by Komatsu that if S() is a strongly continuous semigroup in a Banach space E then the space of all u E such that t S(t)u possesses a fractional derivative of order α 0 coincides with the domain of the α-th power of (a translate of) the infinitesimal generator A. We prove here that a similar relationship holds for strongly continuous cosine functions, at least if E belongs to a class including Hilbert spaces; in general Banach spaces only an inclusion can be assured.

Mathematical Subject Classification
Primary: 47D05, 47D05
Received: 8 May 1981
Revised: 28 May 1982
Published: 1 December 1983
Hector O. Fattorini