Traces of operators with relative trace-class perturbation.
The authors announce results of the following kind obtained in their paper [Sb. Math. 193, No. 2, 279--302 (2002); translation from Mat. Sb. 193, No. 2, 129--152 (2002; Zbl 1033.47012)]: If \(A_0\) is a selfadjoint operator in a separable Hilbert space whose resolvent is of trace class and which has eigenvalues \(\{\lambda_n\}_{n=-\infty}^{\infty}\), and \(B\) a bounded operator, and if \(\{\mu_n\}_{n=-\infty}^{\infty}\) are the eigenvalues of \(A_0+B\), then there exist sequences \(\{n_m\}_{m=1}^{\infty}\) and \(\{k_m\}_{m=1}^{\infty}\) of positive integers such that \[ \lim_{m\rightarrow \infty} \sum_{\ell=-k_m}^{n_m} \bigl(\mu_{\ell}- \lambda_{\ell} -(B \varphi_{\ell}, \varphi_{\ell})\bigr) = 0, \] where \(\{\varphi_n\}\) is the eigenbasis for \(A_0\).
selfadjoint operator with compact resolvent
scientific article; zbMATH DE number 2074829