On diagonalization of certain classes of linear operators (Q1902842)

Let \(A\) be a normal compact operator in a separable complex Hilbert space \(H\), \(\{\lambda_1, \lambda_2,\dots\}\) be the eigenvalues of \(A\), and \(\{e_1,e_2,\dots\}\) be an orthonormal basis in \(H\) consisting of the corresponding eigenvectors of \(A\). Assume that there is a \(\rho\in(0,1)\) such that \(|\lambda_{n+1}|\leq \rho|\lambda_n|\) for all \(n\in\mathbb{N}\). In this paper the question of diagonalizability of the operator \(A-B\) under a certain restriction upon the operator \(B\in L(H)\) is considered. The obtained results improve some recent results of \textit{A. Hinkkanen} [Mich. Math. J. 32, 349-359 (1985; Zbl 0636.47032)].