Cosine families in GDP Quojection-Fréchet spaces
DOI:
https://doi.org/10.5269/bspm.48149Abstract
We prove that if the Quojection-Fréchet space $X$ is a Grothendieck space with the Dunford-Pettis property, then every $C_ {0}$-cosine family is necessarily uniformly continuous and therefore its infinitesimal generator is a continuous linear operator.
References
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2. A. A. Albanese, J. Bonet, W. J. Ricker, C0-semigroups and mean ergodic operators in a class of Frechet spaces, J. Math. Anal. Appl. 365, (2010), 142-157.
3. R. Ameziane, A. Blali, A. Elamrani, K. Moussaouja, Cosine families of operators in a class of Frechet spaces, Proy. J. Math. 37, (2018), 103-118. https://doi.org/10.4067/S0716-09172018000100103
4. W. Arendt, C. J. K. Batty, M. Hieber, F. Neubrander, Vector-valued Laplace transforms and Cauchy problems, Birkhauser, Basel, (2001). https://doi.org/10.1007/978-3-0348-5075-9
5. S. F. Bellenot, E. Dudinsky, Frechet spaces with nuklear Kothe. Trans. Amer. Math. Soc. 237, (1982), 579-594. https://doi.org/10.2307/1999929
6. J. A. Conejero, On the existence of transitive and topologically mixing semigrous. Bull. Bely. Math. soc. Simon stevin 14, (2007), 463-471. https://doi.org/10.36045/bbms/1190994207
7. S. Dierolf, D. N. Zarnadze, A note on strictly regular Frechet spaces, Arch. Math. 42, (1984), 549-556. https://doi.org/10.1007/BF01194053
8. L. Dieter, Strongly continuous operator cosine functions, Functional analysis (Dubrovnik, 1981) Lecture Notes in Math., vol. 948, Springer, Berlin-New York, (1982), 73-97. https://doi.org/10.1007/BFb0069842
9. R. Edwards, Functional Analysis, Reinhart and Winston, New York, 1965.
10. H. O. Fattorini, Second order linear differential equations in Banach spaces, Elsevier Science Publishers B.V, Amsterdam, (1969). https://doi.org/10.1016/0022-0396(69)90105-3
11. L. Frerick, E. Jorda, T. Kalmes, J. Wengenroth, Strongly continuous semigroups on some Fr'echet spaces, J. Math. Ana. Appl. 412, (2014), 121-124. https://doi.org/10.1016/j.jmaa.2013.10.053
12. M. Sova, Cosine operator functions. Rozprawy Mat. 49, (1966), 1-47.
13. K. Yosida, Function analysis, Springer-Verlag, Berlin, (1980).
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2022-01-30
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