Ufa Mathematical Journal
RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB  
General information
Latest issue
Archive
Impact factor

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Ufimsk. Mat. Zh.:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Ufa Mathematical Journal, 2021, Volume 13, Issue 1, Pages 3–16
DOI: https://doi.org/10.13108/2021-13-1-3
(Mi ufa545)
 

This article is cited in 2 scientific papers (total in 2 papers)

On rank one perturbations of semigroup of shifts on half-axis

G. G. Amosovabcd, E. L. Baitenovd

a Steklov Mathematical Institute, RAS, Gubkina str. 8, 119991, Moscow, Russia
b Institute of Mathematics, Ufa Federal Research Center, RAS, Chernyshevsky str. 112, 450008, Ufa, Russia
c St.-Petersburg State University, Universitetstkaya nab. 7-9, 199034, St.-Petersburg, Russia
d Moscow Institute of Physics and Technology, National Research University, Institutskiy per. 9, 141701, Dolgoprudny, Russia
References:
Abstract: We study a special case of perturbations of the semigroup of shifts on the half-axis changing the domain of its generator. We consider a rank one perturbation of generator defined by an exponential. We show that such perturbation of the generator always produces the generator of some $C_0$-semigroup, the action of which is described explicitly. The criterion of isometricity and contractivity of the perturbed semigroup is obtained. For the contractive case, we show that the considered generator perturbation produces a rank one perturbation of the cogenerator. The studied special case is used to build a model of perturbation for the semigroup of shifts defined by an integral equation with respect to some operator-valued measure. In the case when the domain of the generator remains unchanged, this integral equation is reduced to a well-known equation of the perturbation theory, where the integration is made with respect to the usual Lebesgue measure. If the domain is changed, the perturbation satisfies an integral equation with a nontrivial measure that having no density with respect to the Lebesgue measure. We study completely the problem of constructing an operator-valued measure that defines the integral equation relating the perturbed semigroup with the original one. The measure, when it exists, is obtained explicitly and we show that it is defined non-uniquely. We study the possibility of choosing an operator-valued measure with values in the set of self-adjoint and positive operators.
Keywords: semigroup of shifts, rank one perturbations of generator, perturbations changing the domain of generator.
Received: 21.09.2020
Russian version:
Ufimskii Matematicheskii Zhurnal, 2021, Volume 13, Issue 1, Pages 3–16
Bibliographic databases:
Document Type: Article
UDC: 512.98
MSC: 47В06, 46L51
Language: English
Original paper language: Russian
Citation: G. G. Amosov, E. L. Baitenov, “On rank one perturbations of semigroup of shifts on half-axis”, Ufimsk. Mat. Zh., 13:1 (2021), 3–16; Ufa Math. J., 13:1 (2021), 3–16
Citation in format AMSBIB
\Bibitem{AmoBai21}
\by G.~G.~Amosov, E.~L.~Baitenov
\paper On rank one perturbations of semigroup of shifts on half-axis
\jour Ufimsk. Mat. Zh.
\yr 2021
\vol 13
\issue 1
\pages 3--16
\mathnet{http://mi.mathnet.ru/ufa545}
\transl
\jour Ufa Math. J.
\yr 2021
\vol 13
\issue 1
\pages 3--16
\crossref{https://doi.org/10.13108/2021-13-1-3}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000678390800001}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85104055911}
Linking options:
  • https://www.mathnet.ru/eng/ufa545
  • https://doi.org/10.13108/2021-13-1-3
  • https://www.mathnet.ru/eng/ufa/v13/i1/p3
  • This publication is cited in the following 2 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Уфимский математический журнал
    Statistics & downloads:
    Abstract page:316
    Russian version PDF:165
    English version PDF:31
    References:39
     
      Contact us:
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2024