Let A be a compact linear operator in a Banach space X. The author investigates the iterations of a given subspace M of X. He formulates sufficient conditions for the convergence (in the sense of the gap between subspaces). This result is applied to the study of spectra of operators.
This is an interesting expository article about the approximation of operators on a complex infinite-dimensional Hilbert space. Although the article does not include research published during the past twenty years or so, it provides a nice account of the stability of the spectrum under approximation (or perturbation). The reader interested in pursuing this area of research might refer to the bibliography in [D. A. Herrero, Approximation of Hilbert space operators. Vol. I, Pitman, Boston, MA, 1982; MR0676127] and the subsequent publications of those authors.
In a finite-dimensional real or complex linear normed space X there are characterized all the sets of operators A1,...,An which sum up to the identity operator and such that ||Aix||+...+||Anx|=||x|| for all xX. An example of application in the theory of stability of difference schemes is given.
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The aim of this paper is to find estimates of the Hausdorff distance between the spectra of two nonselfadjoint operators. The operators considered are assumed to have their imaginary parts in some normed ideal of compact operators. In the case of the classical Schatten ideals the estimates are given explicitly.
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Table des Matières § 1. Généralités. Matrices infinies............................................ 5 § 2. Systèmes infinis d'équations différentielles. Problème de Cauchy. Existence- unicité............................................................................ 10 § 3. Approximation par les systèmes finis......................................... 20 § 4. Dépendance de la solution des conditions initiales ou d'un paramètre......... 23 § 5. Quelques remarques sur la séparation des variables........................... 25 Les ouvrages cités................................................................. 32
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