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Mathematics of the USSR-Sbornik, 1976, Volume 30, Issue 3, Pages 311–320
DOI: https://doi.org/10.1070/SM1976v030n03ABEH002276
(Mi sm2905)
 

On the imaginary component of a dissipative operator with slowly increasing resolvent

Yu. P. Ginzburg
References:
Abstract: We consider the class $\Lambda$ (RZhMat., 1970, 6B675) of bounded dissipative operators with real spectrum acting in the infinite-dimensional separable Hilbert space $\mathfrak H$ whose resolvents $R_A(\lambda)$ satisfy the following growth condition:
$$ \varlimsup_{y\to+0}\int_{-\infty}^\infty(1+x^2)^{-1}\ln^+y\,\|R_A(x+iy)\|\,dx<\infty. $$
Principal results:
1. The operator $H\geqslant0$ is the imaginary component of an operator $A\in\Lambda$ (i.e., $H=(1/2i)(A-A^*)$) if and only if $0$ is either an eigenvalue of infinite multiplicity for $H$ or a limit point for the spectrum of $H$.
2. All linear operators with imaginary component $H\geqslant0$ and real spectrum belong to the class $\Lambda$ if and only if $H$ is nuclear: $\operatorname{tr}H<\infty$.
Bibliography: 10 titles.
Received: 30.12.1974
Bibliographic databases:
UDC: 519.56+513.88
MSC: Primary 47B44; Secondary 47B10
Language: English
Original paper language: Russian
Citation: Yu. P. Ginzburg, “On the imaginary component of a dissipative operator with slowly increasing resolvent”, Math. USSR-Sb., 30:3 (1976), 311–320
Citation in format AMSBIB
\Bibitem{Gin76}
\by Yu.~P.~Ginzburg
\paper On the imaginary component of a~dissipative operator with slowly increasing resolvent
\jour Math. USSR-Sb.
\yr 1976
\vol 30
\issue 3
\pages 311--320
\mathnet{http://mi.mathnet.ru//eng/sm2905}
\crossref{https://doi.org/10.1070/SM1976v030n03ABEH002276}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=440406}
\zmath{https://zbmath.org/?q=an:0355.47017}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=A1976FN58700003}
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  • https://doi.org/10.1070/SM1976v030n03ABEH002276
  • https://www.mathnet.ru/eng/sm/v143/i3/p349
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