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 Mat. Sb. (N.S.), 1976, Volume 101(143), Number 3(11), Pages 349–359 (Mi msb2905)

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

Yu. P. Ginzburg

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.

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English version:
Mathematics of the USSR-Sbornik, 1976, 30:3, 311–320

Bibliographic databases:

UDC: 519.56+513.88
MSC: Primary 47B44; Secondary 47B10

Citation: Yu. P. Ginzburg, “On the imaginary component of a dissipative operator with slowly increasing resolvent”, Mat. Sb. (N.S.), 101(143):3(11) (1976), 349–359; 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 Mat. Sb. (N.S.) \yr 1976 \vol 101(143) \issue 3(11) \pages 349--359 \mathnet{http://mi.mathnet.ru/msb2905} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=440406} \zmath{https://zbmath.org/?q=an:0355.47017} \transl \jour Math. USSR-Sb. \yr 1976 \vol 30 \issue 3 \pages 311--320 \crossref{https://doi.org/10.1070/SM1976v030n03ABEH002276} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1976FN58700003}