On the compactness of strongly continuous semigroups and cosine functions of operators

Henríquez, Hernán R.

doi:10.1090/S0002-9939-1995-1227517-2

On the compactness of strongly continuous semigroups and cosine functions of operators
HTML articles powered by AMS MathViewer

by Hernán R. Henríquez

Proc. Amer. Math. Soc. 123 (1995), 1417-1424

DOI: https://doi.org/10.1090/S0002-9939-1995-1227517-2

PDF | Request permission

Abstract:

In this note we relate two notions of compactness for strongly continuous semigroups of linear operators and cosine functions of linear operators. Specifically, if T denotes a strongly continuous semigroup of linear operators defined on a Banach space X, we will show that T is compact if and only if the set $\{ (T( \bullet )x:x \in X,\left \| x \right \| \leq 1\}$ is relatively compact in any space ${L^p}([0,a]);X)$ for $1 \leq p < \infty$ and $a > 0$. We establish similar results for ${(T(t) - I)^n},n \in {\mathbf {N}}$, and for cosine and sine functions of operators.

References

Klaus Deimling, Nonlinear functional analysis, Springer-Verlag, Berlin, 1985. MR 787404, DOI 10.1007/978-3-662-00547-7
H. O. Fattorini, Second order linear differential equations in Banach spaces, North-Holland Mathematics Studies, vol. 108, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 99. MR 797071

Uma propriedade compacidade para familias cosseno de operadores

Una nota sobre la compacidad de funciones coseno de operadores

6

Hernán R. Henríquez, Periodic solutions of quasi-linear partial functional-differential equations with unbounded delay, Funkcial. Ekvac. 37 (1994), no. 2, 329–343. MR 1299869
James H. Lightbourne III, Periodic solutions and perturbed semigroups of linear operators, Nonlinear systems and applications (Proc. Internat. Conf., Univ. Texas, Arlington, Tex., 1976) Academic Press, New York, 1977, pp. 591–602. MR 0458263
Charles-Michel Marle, Mesures et probabilités, Collection Enseignement des Sciences, No. 19, Hermann, Paris, 1974 (French). MR 0486378
W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, R. Nagel, F. Neubrander, and U. Schlotterbeck, One-parameter semigroups of positive operators, Lecture Notes in Mathematics, vol. 1184, Springer-Verlag, Berlin, 1986. MR 839450, DOI 10.1007/BFb0074922
A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486, DOI 10.1007/978-1-4612-5561-1
Angus E. Taylor, Introduction to functional analysis, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1958. MR 0098966
C. C. Travis and G. F. Webb, Second order differential equations in Banach space, Nonlinear equations in abstract spaces (Proc. Internat. Sympos., Univ. Texas, Arlington, Tex., 1977) Academic Press, New York, 1978, pp. 331–361. MR 502551
C. C. Travis and G. F. Webb, Compactness, regularity, and uniform continuity properties of strongly continuous cosine families, Houston J. Math. 3 (1977), no. 4, 555–567. MR 500288
I. I. Vrabie, Compactness methods for nonlinear evolutions, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 32, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1987. With a foreword by A. Pazy. MR 932730
L. W. Weis, A generalization of the Vidav-Jörgens perturbation theorem for semigroups and its application to transport theory, J. Math. Anal. Appl. 129 (1988), no. 1, 6–23. MR 921374, DOI 10.1016/0022-247X(88)90230-2