Abstract
This paper is concerned first with the behaviour of differences T(t)-T(s) near the origin, where (T(t)) is a semigroup of operators on a Banach space, defined either on the positive real line or a sector in the right half-plane (in which case it is assumed analytic). For the non-quasinilpotent case extensions of results in the published literature are provided, with best possible constants; in the case of quasinilpotent semigroups on the half-plane, it is shown that, in general, differences such as T(t)-T(2t) have norm approaching 2 near the origin. The techniques given enable one to derive estimates of other functions of the generator of the semigroup; in particular, conditions are given on the derivatives near the origin to guarantee that the semigroup generates a unital algebra and has bounded generator.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bendaoud, Z., Esterle, J., Mokhtari, A.: Distances entre exponentielles et puissances d’éléments de certaines algèbres de Banach. Arch. Math. (Basel), 89(3), 243–253 (2007)
Esterle, J., Mokhtari, A.: Distance entre éléments d’un semi-groupe dans une algèbre de Banach. J. Funct. Anal., 195(1), 167–189 (2002)
Hille, E.: On the differentiability of semi-group operators. Acta Sci. Math. (Szeged), 12, 19–24 (1950)
Hille, E., Phillips, R. S.: Functional Analysis and Semi-groups, American Mathematical Society, Vol. XXXI, Third printing of the revised edition of 1957, American Mathematical Society, Providence, 1974
Rickart, C. E.: General Theory of Banach Algebras, The University Series in Higher Mathematics. D. van Nostrand Co., Inc., Princeton, Toronto-London-New York, 1960
Halmos, P. R.: Measure Theory, D. van Nostrand Company, Inc., New York, 1950
Beurling, A.: On analytic extension of semigroups of operators. J. Funct. Anal., 6, 387–400 (1970)
Chalendar, I., Esterle, J., Partington J. R.: Boundary values of analytic semigroups and associated norm estimates. Preprint
Feller, W.: On the generation of unbounded semi-groups of bounded linear operators. Ann. of Math., 58(2), 166–174, 1953
Kalton, N., Montgomery-Smith, S., Oleszkiewicz, K., Tomilov, Y.: Power-bounded operators and related norm estimates. J. Lond. Math. Soc., 70(2), 463–478 (2004)
Esterle, J.: Distance near the origin between elements of a strongly continuous semigroup. Ark. Mat., 43(2), 365–382 (2005)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by EPSRC (EP/F020341/1); the third author is partially supported by the research project AHPI, funded by ANR
Rights and permissions
About this article
Cite this article
Bendouad, Z., Chalendar, I., Esterle, J. et al. Distances between elements of a semigroup and estimates for derivatives. Acta. Math. Sin.-English Ser. 26, 2239–2254 (2010). https://doi.org/10.1007/s10114-010-8569-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-010-8569-6