Regularized traces of nonself-adjoint operators (Q1974369)

Let \(T\) be a semibounded selfadjoint operator with trace-class resolvent, \(\lambda _n\) the nondecreasing eigenvalues of \(T\), and \(v_n\) the corresponding orthonormal eigenvectors. Next, let \(P\) be a bounded operator, and let \(\mu _n\) be the eigenvalues of \(T+P\) indexed counting algebraic multiplicities and in such a way that the \(\Re \mu _n\) increase. If the numbers \(\lambda _n\) behave as \(cn^{\alpha}\), then \[ \lim_{m\to\infty}\sum^{n_m}_{n=1}[\mu _n -\lambda _n -(Pv_n, v_n)] =0, \] where \(n_m\) are arbitrary positive integers satisfying \(\lambda _{n_{m+1}} -\lambda _{n_m} \geq 2^{-1}cn_m ^{\alpha -1}\) (such numbers \(n_m\) always exist). An application to differential operators is discussed briefly.