For generators of n-parameter strongly continuous operator semigroups in a Banach space, we construct a Hille-Phillips type functional calculus, the symbol class of which consists of analytic functions from the image of the Laplace transform of the convolution algebra of temperate distributions supported by the positive cone ℝ n+ . The image of such a calculus is described with the help of the commutant of the semigroup of shifts along the cone. The differential properties of the calculus and some examples are presented.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Hille E. and Phillips R. S., Functional Analysis and Semigroups, Amer. Math. Soc., Providence (1957).
Nelson E. A., “Functional calculus using singular Laplace integrals,” Trans. Amer. Math. Soc., 88, 400–413 (1958).
Balakrishnan A. V., “An operational calculus for infinitesimal operators of semigroups,” Trans. Amer. Math. Soc., 91, 330–353 (1960).
Mirotin A. R., “On the T -calculus of generators for C 0-semigroups,” Siberian Math. J., 39, No. 3, 493–503 (1998).
Mirotin A. R., “On some functions that send each generator of a C 0-semigroup to the generator of a holomorphic semigroup,” Siberian Math. J., 43, No. 1, 114–123 (2002).
Mirotin A. R., “On some properties of the multidimensional Bochner-Phillips functional calculus,” Siberian Math. J., 52, No. 6, 1032–1041 (2011).
Baeumer B., Haase M., and Kovács M., “Unbounded functional calculus for bounded groups with applications,” J. Evol. Equ., 9, No. 1, 171–195 (2009).
Butzer P. L. and Berens H., Semi-Groups of Operators and Approximation, Springer-Verlag, Berlin, Heidelberg, and New York (1967).
Vladimirov V. S., Generalized Functions in Mathematical Physics [in Russian], Nauka, Moscow (1979).
Zharinov V. V., “Compact families of locally convex topological vector spaces, Fréchet-Schwartz and dual Fréchet-Schwartz spaces,” Russian Math. Surveys, 34, No. 4, 105–143 (1979).
Komatsu H., An Introduction to the Theory of Generalized Functions, Tokyo Univ. Publ., Tokyo (2000).
Seeley R. T., “Extensions of C ∞-functions defined in a half-space,” Proc. Amer. Math. Soc., 15, 625–626 (1964).
Schaefer H. H., Topological Vector Spaces, Springer-Verlag, New York, Heidelberg, and Berlin (1971).
Schwartz L., “Espaces de fonctions différentielles à valeurs vectorielles,” J. Anal. Math., 4, 88–148 (1954/55).
Raĭkov D. A., “Double closed-graph theorem for topological linear spaces,” Siberian Math. J., 7, No. 2, 287–300 (1967).
Original Russian Text Copyright © 2014 Lopushansky O.V. and Sharyn S.V.
Rzeszow and Ivano-Frankivsk. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 55, No. 1, pp. 131–146, January–February, 2014.
About this article
Cite this article
Lopushansky, O.V., Sharyn, S.V. Generalized Hille-Phillips type functional calculus for multiparameter semigroups. Sib Math J 55, 105–117 (2014). https://doi.org/10.1134/S0037446614010133
- Hille-Phillips calculus
- temperate distribution
- Laplace transform