Some applications of Fejer's theorem to operator cosine functions in Banach spaces

Authors:
Ioana Cioranescu and Carlos Lizama

Journal:
Proc. Amer. Math. Soc. **125** (1997), 2353-2362

MSC (1991):
Primary 47D09; Secondary 34G10

DOI:
https://doi.org/10.1090/S0002-9939-97-03837-9

MathSciNet review:
1389510

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Abstract | References | Similar Articles | Additional Information

Abstract: We characterize spectral properties of operator cosine functions in Banach spaces in terms of the Césaro summability of two series associated to the resolvent of the corresponding infinitesimal generator.

**1.**I. Cioranescu, C. Lizama,*Spectral properties of cosine operator functions*, Aequationes Math. 36 (1988) 80-98. MR**89i:47071****2.**V.I. Gorbachuk, A.V. Knyazyuk,*Boundary values of solutions of operator-differential equations*, Russian Math. Surveys 44:3 (1989) 67-111. MR**91c:47080****3.**G. Greiner,*A short proof of Gearhart's theorem*, Semesterbericht Funktionalanalysis Tubingen SS 89 16 (1989) 89-92.**4.**G. Greiner, M. Schwarz,*Weak spectral mapping theorems for functional differential equations*, J.Diff. Eq. 94 (1991) 205-216. MR**92k:47079****5.**Y. Katznelson, Harmonic Analysis, New York-London-Sydney-Toronto, 1968. MR**40:1734****6.**Y. Latushkin, S. Montgomery-Smith,*Lyapunov theorems for Banach spaces*, Bull. Amer. Math. Soc. (New series) 31 (1994) 44-49. MR**94j:47062****7.**C. Lizama,*On the spectrum of cosine operator functions*, Int. Eq. Operator Theory 12 (1989) 713-724. MR**90h:47074****8.**R.Nagel,*One-Parameter Semigroups of Positive Operators*, Lecture Notes in Mathematics, Vol. 1184, Springer-Verlag, Berlin/New York, 1980. MR**88i:47022****9.**B. Nagy,*On cosine operator functions in Banach spaces*, Acta Sci. Math. (Szeged) 36 (1974) 281-290. MR**51:11191****10.**C.C. Travis, G.F. Webb,*Cosine families and abstract non-linear second order differential equations*, Acta Math. Acad. Sci. Hung. 32 (1978) 75-96. MR**58:17404**

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Additional Information

**Ioana Cioranescu**

Affiliation:
Department of Mathematics, University of Puerto Rico, Box 23355 Rio Piedras, Puerto Rico 00931

**Carlos Lizama**

Affiliation:
Departamento de Matemática y C. C., Universidad de Santiago de Chile, Casilla 307-Correo 2, Santiago, Chile

Email:
clizama@fermat.usach.cl

DOI:
https://doi.org/10.1090/S0002-9939-97-03837-9

Received by editor(s):
December 11, 1995

Received by editor(s) in revised form:
February 22, 1996

Additional Notes:
This research was done while the first author was visiting the Department of Mathematics, University of Santiago de Chile, supported by CONICYT

The second author was partially supported by FONDECYT grant 1930066 and DICYT (USACH)

Communicated by:
Palle E. T. Jorgensen

Article copyright:
© Copyright 1997
American Mathematical Society