Cosine families of operators in a class of Fréchet spaces.
Keywords:
Strongly continuous cosine families, emi-groups of operators, Locally convex spac, Quojection Fréchet spaceAbstract
M. sova [10] proved that the infinitesimal generator of all uniformly continuous cosine family, of operators in Banach space, is a bounded operator. We show by counter-example that the result mentioned above is not true in general on Fréchet spaces, and we prove that the infinitesimal generator of an uniformly continuous cosine family of operators in a class of Fréchet spaces (quojection) is necessarily continuous.
References
Albanese, A. A., Bonet, J. Ricker, W. J., Grothendieck spaces with the DunfordPettis property, Positivity, 14, pp. 145-164, (2010).
Arendt, W., Batty, C. J. K., Hieber, M., Neubrander, F.: VectorValued Laplace Transforms and Cauchy Problems, Birkhauser, Basel, pp. 202-210, (2001)
Dierolf, S., Zarnadze, D. N., A note on strictly regular Fréchet spaces, Arch. Math., 42, pp. 549-556, (1984).
Dieter L., Strongly continuous operator cosine functions, Functional analysis (Dubrovnik, 1981) Lecture Notes in Math., vol. 948, Springer, Berlin-New York, pp. 73-97 (1982)
Fattorini, H. O.: Ordinary differential equations in linear topological spaces, II. J. Differential Equations 6, pp. 24-34, (1969)
Jarchow, H., locally Convex Space, pp. 156, B. G. Teubner, stuttgart, (1981)
Meise, R., Vogt, D, introduction to Function Analysis, pp.326-356, Clarendon press, Oxford, (1997)
Schaefer, H. H, Topological Vector Spaces, pp. 101. Graduate textes in MAthematics, springer-verlag, Berlin Heidelberg New York, (1971)
Serge L., Real and Functional Analysis, pp.396. third ed., Grad. Texts in Math., Vol. 142, Springer-Verlag, NewYork, (1993)
Sova, M,. Cosine operator functions. Rozprawy Mat. 49, pp. 1-47, (1966)
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