Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)


For citation:

Strukov V. E., Strukova I. I. Harmonic Analysis of Operator Semigroups Slowly Varying at Infinity. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2019, vol. 19, iss. 2, pp. 152-163. DOI: 10.18500/1816-9791-2019-19-2-152-163, EDN: UHXOAR

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
28.05.2019
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Russian
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Article
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517.9
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UHXOAR

Harmonic Analysis of Operator Semigroups Slowly Varying at Infinity

Autors: 
Strukov Victor Evgen'evich, Voronezh State University
Strukova Irina Igorevna, Voronezh State University
Abstract: 

The article focuses on studying of strongly continuous bounded operator semigroups. In the space of uniformly continuous functions with values inacomplex Banach space weconsider the subspace of integrally vanishing at infinity functions. This subspace includes the subspace of vanishing at infinity functions, but it is wider. We study the properties of the subspace under consideration. We introduce the definition of slowly varying at infinity (with regard to the subspace of integrally vanishing at infinity functions) function and study the conditions under which a uniformly continuous function belongs to this type. We also introduce the definition of slowly varying at infinity (with regard to the subspace of integrally vanishing at infinity functions) operator semigroup, study its properties and derive the condition sunder which a strongly continuous bounded operator semigroup belongs to this type. The results derived in the article might be useful for research of stabilization of parabolic equations solutions with unlimited increase of time.

References: 
  1. Baskakov A.G., Kaluzhina N.S. Beurling’s theorem for functions with essential spectrum from homogeneous spaces and stabilization of solutions of parabolic equations. Math. Notes, 2012, vol. 92, no. 5, pp. 587–605. DOI: https://doi.org/10.1134/S0001434612110016
  2. Strukova I. I. Spectra of algebras of slowly varying and periodic at infinity functions and Banach limits. Proceedings of Voronezh State University. Ser. Physics. Mathematics, 2015, no. 3, pp. 161–165 (in Russian).
  3. Baskakov A., Strukova I. Harmonic analysis of functions periodic at infinity. Eurasian Math. J., 2016, vol. 7, no. 4, pp. 9–29.
  4. Strukova I. I. About Wiener theorem for periodic at infinity functions. Siberian Math. J., 2016, vol. 57, iss. 1, pp. 145–154. DOI: https://doi.org/10.1134/S0037446616010146
  5. Strukova I. I. About harmonic analysis of periodic at infinity functions. Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 2014, vol. 14, iss. 1, pp. 28–38 (in Russian).
  6. Strukova I. I. Harmonic analysis of periodic at infinity functions from Stepanov spaces. Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 2017, vol. 17, iss. 2, pp. 172–182 (in Russian). DOI: https://doi.org/10.18500/1816-9791-2017-17-2-172-182
  7. Baskakov A. G., Kaluzhina N. S., Polyakov D. M. Slowly varying at infinity operator semigroups. Russian Math. (Iz. VUZ), 2014, vol. 58, no. 7, pp. 1–10. DOI: https://doi.org/10.3103/S1066369X14070019
  8. Baskakov A. G. Theory of representations of Banach algebras, and abelian groups and semigroups in the spectral analysis of linear operators. J. Math. Sci. (N. Y.), 2006, vol. 137, no. 4. pp. 4885–5036. DOI: https://doi.org/10.1007%2Fs10958-006-0286-4
  9. Baskakov A. G., Krishtal I. A. Harmonic analysis of causal operators and their spectral properties. Izv. Math., 2005, vol. 69, no. 3, pp. 439–486. DOI: https://doi.org/10.1070/IM2005v069n03ABEH000535
  10. Baskakov A. G. Garmonicheskij analiz linejnykh operatorov [Harmonic analysis of linear operators]. Voronezh, VSU Publ., 1987. 165 p. (in Russian).
  11. Baskakov A. G. Spectral tests for the almost periodicity of the solutions of functional equations. Math. Notes, 1978, vol. 24, no. 1–2, pp. 606–612. DOI: https://doi.org/10.1007/BF01105312
  12. Baskakov A. G. Bernˇ ste˘ ın-type inequalities in abstract harmonic analysis. Siberian Math. J., 1979, vol. 20, no. 5, pp. 665–672.
  13. Wiener N. The Fourier Integral and Certain of its Applications. Cambridge Univ. Press, reprint by Dover, CUP Archive, 1988, 201 p. (Russ. ed.: Moscow, Fizmatlit, 1963. 256 p.).
  14. Chicone C., Latushkin Y. Evolution Semigroups in Dynamical Systems and Differential Equations. Amer. Math. Soc., 1999, vol. 70, 361 p.
  15. Baskakov A. G. Semigroups of difference operators in spectral analysis of linear differential operators. Funct. Anal. Its Appl., 1996, vol. 30, iss. 3. pp. 149–157. DOI: https://doi.org/10.1007/BF02509501
  16. Baskakov A. G. Linear differential operators with unbounded operator coefficients and semigroups of bounded operators. Math. Notes, 1996, vol. 59, no. 6, pp. 586–593. DOI: https://doi.org/10.1007/BF02307207
  17. Baskakov A. G. Spectral analysis of differential operators with unbounded operator-valued coefficients, difference relations and semigroups of difference relations. Izv. Math., 2009, vol. 73, no. 2, pp. 215–278. DOI: https://doi.org/10.1070/IM2009v073n02ABEH002445
  18. Baskakov A. G. Analysis of linear differential equations by methods of the spectral theory of difference operators and linear relations. Russian Math. Surveys, 2013, vol. 68, no. 1, pp. 69–116. DOI: https://doi.org/10.1070/RM2013v068n01ABEH004822
  19. Baskakov A. G. Harmonic and spectral analysis of power bounded operators and bounded semigroups of operators on Banach spaces. Math. Notes, 2015, vol. 97, no. 2, pp. 164–178. DOI: https://doi.org/10.1134/S0001434615010198
Received: 
05.05.2018
Accepted: 
03.02.2019
Published: 
28.05.2019
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