Analytic properties of cosine operators

Nelson, S.; Triggiani, R.

doi:10.1090/S0002-9939-1979-0521880-4

Analytic properties of cosine operators
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by S. Nelson and R. Triggiani PDF

Proc. Amer. Math. Soc. 74 (1979), 101-104 Request permission

Abstract:

Let A be the infinitesimal generator of a strongly continuous cosine operator $C(t)$, hence of the analytic semigroup $S(t)$, on the Banach space X. It is proved that the set of analytic vectors for $C(t)$ contains the dense subspace ${X_0} = { \cup _{0 < t}}S(t)X$, the containment being in general proper.

References

Semigroups of operators and approximations

H. O. Fattorini, Ordinary differential equations in linear topological spaces. I, J. Differential Equations 5 (1969), 72–105. MR 277860, DOI 10.1016/0022-0396(69)90105-3

Functional analysis and semigroups

J. Kisyński, On cosine operator functions and one-parameter groups of operators, Studia Math. 44 (1972), 93–105. MR 312328, DOI 10.4064/sm-44-1-93-105
J. Kisyński, On operator-valued solutions of d’Alembert’s functional equation. I, Colloq. Math. 23 (1971), 107–114. MR 313875, DOI 10.4064/cm-23-1-107-114
Svetozar Kurepa, A cosine functional equation in Banach algebras, Acta Sci. Math. (Szeged) 23 (1962), 255–267. MR 145370
Raymond E. A. C. Paley and Norbert Wiener, Fourier transforms in the complex domain, American Mathematical Society Colloquium Publications, vol. 19, American Mathematical Society, Providence, RI, 1987. Reprint of the 1934 original. MR 1451142, DOI 10.1090/coll/019
M. Sova, Cosine operator functions, Rozprawy Mat. 49 (1966), 47. MR 193525
Roberto Triggiani, On the relationship between first and second order controllable systems in Banach spaces, SIAM J. Control Optim. 16 (1978), no. 6, 847–859. MR 509455, DOI 10.1137/0316058