Regularized traces of a bounded perturbation of an operator with trace class resolvent (Q2704030)

If \(A\) is a trace class operator in a separable Hilbert space, it holds that NEWLINE\[NEWLINE\sum^\infty_{n=1} (A\varphi_n,\varphi_n)= \sum^\infty_{n=1} (A\psi_n, \psi_n)NEWLINE\]NEWLINE for every pair of orthonormal bases \(\{\varphi_n\}^\infty_{n=1}\) and \(\{\psi_n\}^\infty_{n=1}\). In the paper under review the authors consider the problem in a little more general situation, for operators which may not be of trace class, to ask when a modified relation NEWLINE\[NEWLINE\sum^\infty_{n=1} [(A\varphi_n, \varphi_n)- (A\psi_n, \psi_n)]= 0NEWLINE\]NEWLINE holds instead. Indeed, each of \(\sum^\infty_{n=1} (A\varphi_n, \varphi_n)\) and \(\sum^\infty_{n=1} (A\psi_n, \psi_n)\) may possibly diverge, but the above modified sum may converge because of cancellation.NEWLINENEWLINENEWLINEThe main result is: Let \(A_0\) be a selfadjoint operator with compact resolvent whose eigenvalues \(\{\lambda_n\}^\infty_{n= -\infty}\) satisfy \(\sum^\infty_{n=-\infty} 1/|\lambda_n|< \infty\) and also some additional conditions of growth order related to \(n\). With \(B\) a bounded linear operator, let \(\{\mu_n\}^\infty_{n=-\infty}\) be the eigenvalues of \(A_0+ B\). Then there exist sequences \(\{n_m\}^\infty_{m=1}\) and \(\{k_m\}^\infty_{m=1}\) of positive integers such that NEWLINE\[NEWLINE\lim_{m\to \infty} \Biggl[\sum^{n_m}_{\ell=- k_m} (\mu_\ell- \lambda_\ell)- (B\varphi_\ell, \varphi_\ell)\Biggr]= 0.NEWLINE\]NEWLINE Here \(\{\varphi_\ell\}^\infty_{\ell= -\infty}\) are the eigenvectors of \(A_0\) corresponding to the eigenvalues \(\{\lambda_n\}^\infty_{n= -\infty}\).NEWLINENEWLINENEWLINETo get this result, they observe that the trace norm of the resolvent \(R_0(\lambda):= (A_0- \lambda I)^{-1}\) converges to zero along a sequence \(\{a_m\}\) of positive real numbers with \(a_m\to \infty\) as \(m\to\infty\), uniformly for \(|\lambda|= a_m\).