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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.
Received: 30.12.1974
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
Linking options:
https://www.mathnet.ru/eng/sm2905https://doi.org/10.1070/SM1976v030n03ABEH002276 https://www.mathnet.ru/eng/sm/v143/i3/p349
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Abstract page: | 245 | Russian version PDF: | 79 | English version PDF: | 4 | References: | 41 |
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