Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Some inequalities for sub-Markovian generators and their applications to the perturbation theory

Authors: V. A. Liskevich and Yu. A. Semenov
Journal: Proc. Amer. Math. Soc. 119 (1993), 1171-1177
MSC: Primary 47D07; Secondary 47A55, 47B25, 47F05
MathSciNet review: 1160303
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We characterize the domain in $ {L^p}$-space of generators of submarkovian semigroups in terms of the form domain in $ {L^2}$ and give the corresponding inequality. Using this inequality we obtain a criterion for the formal difference $ A - B$ of such generators to be a generator of a contraction semigroup in $ {L^p}$. The conditions on perturbation are expressed in terms of forms, i.e., in $ {L^2}$-terms.

References [Enhancements On Off] (What's this?)

  • [CKuS] E. A. Carlen, S. Kusuoka, and D. W. Stroock, Upper bounds for symmetric Markov transition functions, Ann. Inst. H. Poincaré Probab. Statist. 23 (1987), no. 2, suppl., 245–287 (English, with French summary). MR 898496
  • [K] Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
  • [KoSe] V. F. Kovalenko and Yu. A. Semenov, On the 𝐿^{𝑝}-theory of Schrödinger semigroups. I, Ukrain. Mat. Zh. 41 (1989), no. 2, 273–278, 289 (Russian); English transl., Ukrainian Math. J. 41 (1989), no. 2, 246–249. MR 992836,
  • [LPSe] V. A. Liskevich, M. A. Perelmuter, and Yu. A. Semenov, Form-bounded perturbations of generators of submarkovian semigroups (in preparation).
  • [LSe] V. A. Liskevich and Yu. A. Semenov, One criterion of the essential self-adjointness of the Schrödinger operator, preprint, 1991.
  • [RSi] Michael Reed and Barry Simon, Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0493420
  • [St] Elias M. Stein, Topics in harmonic analysis related to the Littlewood-Paley theory., Annals of Mathematics Studies, No. 63, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1970. MR 0252961
  • [S] D. W. Stroock, An introduction to the theory of large deviations, Universitext, Springer-Verlag, New York, 1984. MR 755154
  • [V] N. Th. Varopoulos, Hardy-Littlewood theory for semigroups, J. Funct. Anal. 63 (1985), no. 2, 240–260. MR 803094,

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47D07, 47A55, 47B25, 47F05

Retrieve articles in all journals with MSC: 47D07, 47A55, 47B25, 47F05

Additional Information

Keywords: Submarkovian semigroups, form-bounded perturbations
Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society