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A regularized trace of a bounded perturbation of an operator with a trace-class resolvent. (English. Russian original) Zbl 1056.47504

Dokl. Math. 62, No. 1, 19-21 (2000); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 373, No. 1, 26-28 (2000).
Summary: It was conjectured in [V. A. Sadovnichij and V. E. Podol’skij, Differ. Equations 35, 557–566 (1999; Zbl 0986.47015)] that the formula \[ \lim_{m\to\infty}\, \Biggl(\sum^{n_m}_{l=-k_m} (\mu_l- \lambda_l- (B\varphi_l, \varphi_l))\Biggr)= 0, \] where \(\{\varphi_n\}\) is the basis consisting of the eigenvectors of an operator \(A_0\) with eigenvalues \(\{\lambda_n\}\) and \(\{\mu_n\}\) are the eigenvalues of the operator \(A_0+ B\), holds if \(A_0\) has a nuclear resolvent and if \(B\) is bounded.
In this paper, we give a positive solution to this problem.

MSC:

47A55 Perturbation theory of linear operators
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
47A10 Spectrum, resolvent

Citations:

Zbl 0986.47015
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