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Zeitschrift für Analysis und ihre Anwendungen

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Volume 26, Issue 4, 2007, pp. 391–406
DOI: 10.4171/ZAA/1331

Published online: 2007-12-31

Multiplicative Perturbation by Contractions and Uniform Stability

P. C. M. Vieira[1] and C. S. Kubrusly[2]

(1) National Laboratory for Scientific Computation, Petropolis, Brazil
(2) Catholic University of Rio de Janeiro, Brazil

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).

Keywords: Hilbert space contractions, perturbation theory, spectral radius, uniform stability

Vieira P. C. M., Kubrusly C. S.: Multiplicative Perturbation by Contractions and Uniform Stability. Z. Anal. Anwend. 26 (2007), 391-406. doi: 10.4171/ZAA/1331