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Doklady Rossijskoj Akademii Nauk. Mathematika, Informatika, Processy Upravlenia, 2023, Volume 509, Pages 54–59
DOI: https://doi.org/10.31857/S2686954322600574
(Mi danma361)
 

This article is cited in 1 scientific paper (total in 1 paper)

MATHEMATICS

To Birman–Krein–Vishik theory

M. Malamudab

a Peoples’ Friendship University of Russia, Moscow, Russia
b St. Petersburg State University, St. Petersburg, Russia
Citations (1)
References:
Abstract: Let $A\ge m_A>0$ be a closed positive definite symmetric operator in a Hilbert space $\mathcal H$, let $\hat{A}_F$ and $\hat{A}_K$ be its Friedrichs and Krein extensions, and let $\mathfrak S_\infty$ be the ideal of compact operators in $\mathcal H$. The following problem has been posed by M.S. Birman: Is the implication $A^{-1}\in\mathfrak S_\infty\Rightarrow (\hat{A}_F)^{-1}\in\mathfrak S_\infty(\mathcal H)$ holds true or not? It turns out that under condition $A^{-1}\in\mathfrak{S}_\infty$ the spectrum of Friedrichs extension $\hat{A}_F$ might be of arbitrary nature. This gives a complete negative solution to the Birman problem. Let $\hat{A}'_K$ be the reduced Krein extension. It is shown that certain spectral properties of the operators $(I_{\mathfrak{M}_0}+\hat{A}'_K)^{-1}$ and $P_1(I+A)^{-1}$ are close. For instance, these operators belong to a symmetrically normed ideal $\mathfrak S$, say are compact, only simultaneously. Moreover, it turns out that under a certain additional condition the eigenvalues of these operators have the same asymptotic. Besides we complete certain investigations by Birman and Grubb regarding the equivalence of semiboubdedness property of selfadjoint extensions of $A$ and the corresponding boundary operators.
Keywords: positive definite symmetric operator, Friedrichs and Krein extensions, compactness of resolvent, asymptotic of spectrum.
Funding agency Grant number
Ministry of Science and Higher Education of the Russian Federation 075-15-2021-602
This work is supported by the Ministry of Science and Higher Education of the Russian Federation, agreement 075-15-2021-602.
Presented: S. V. Kislyakov
Received: 07.09.2022
Revised: 16.11.2022
Accepted: 26.12.2022
English version:
Doklady Mathematics, 2023, Volume 107, Issue 1, Pages 44–48
DOI: https://doi.org/10.1134/S1064562423700485
Bibliographic databases:
Document Type: Article
UDC: 517.984
Language: Russian
Citation: M. Malamud, “To Birman–Krein–Vishik theory”, Dokl. RAN. Math. Inf. Proc. Upr., 509 (2023), 54–59; Dokl. Math., 107:1 (2023), 44–48
Citation in format AMSBIB
\Bibitem{Mal23}
\by M.~Malamud
\paper To Birman--Krein--Vishik theory
\jour Dokl. RAN. Math. Inf. Proc. Upr.
\yr 2023
\vol 509
\pages 54--59
\mathnet{http://mi.mathnet.ru/danma361}
\crossref{https://doi.org/10.31857/S2686954322600574}
\elib{https://elibrary.ru/item.asp?id=50436203}
\transl
\jour Dokl. Math.
\yr 2023
\vol 107
\issue 1
\pages 44--48
\crossref{https://doi.org/10.1134/S1064562423700485}
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  • This publication is cited in the following 1 articles:
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    Doklady Rossijskoj Akademii Nauk. Mathematika, Informatika, Processy Upravlenia Doklady Rossijskoj Akademii Nauk. Mathematika, Informatika, Processy Upravlenia
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