Multiplicative Perturbation by Contractions and Uniform Stability

Abstract

Let $T$ be an arbitrary bounded linear transformation of a Hilbert space into itself. We investigate classes of contractions $S$ for which the spectral radius $r(ST)$ of the product $ST$ is less than one. The main result gives a collection of necessary and sufficient conditions for $r(ST)<1$ when $T$ is multiplicatively perturbed by compact contractions $S$. We also give either necessary or sufficient conditions for perturbation by other classes of Hilbert space contractions, such as those that include the symmetries (e.g., involutions, unitary operators, self-adjoint, normal and normaloid contractions) or the orthogonal projections (e.g., nonnegative contractions).