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Stable linear maps

Part of: Normed linear spaces and Banach spaces; Banach lattices General theory of linear operators

Published online by Cambridge University Press:  26 February 2010

M. C. Irwin
M. C. Irwin
Affiliation:
Department of Pure Mathematics, University of Liverpool.
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Extract

Let ˜ be an equivalence relation on a topological space X. A point x ε X i s stable with respect to ˜ if it is in the interior of an equivalence class. We may also add, if ambiguity arises, that x is stable under perturbations in X. Let E be a Banach space, and let L(E) be the Banach space of continuous linear endomorphisms of E, with norm given by |T| = sup{ |T(x) | : |x| = 1}. In this paper we discuss stability of elements of L(E) with respect to some natural equivalence relations.

MSC classification

Secondary: 47A99: None of the above, but in this section 46B99: None of the above, but in this section
Type
Research Article
Information
Mathematika , Volume 18 , Issue 2 , December 1971 , pp. 276 - 282
Copyright
Copyright © University College London 1971

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References

Dunford, N. and Schwartz, J. T., Linear operators (Interscience, New York, 1958).Google Scholar
Hartman, P., Ordinary differential equations (Wiley, New York, 1964).Google Scholar
Holmes, R. B., “A formula for the spectral radius of an operator ”, Amer. Math. Monthly 75, (1968), 163–166.CrossRefGoogle Scholar
Irwin, M. C., “On the smoothness of the composition map ”, Quart. J. Math. (to appear).Google Scholar
Kuiper, N. H., “The homotopy type of the unitary group of Hilbert space ”, Topology, 3 (1965), 19–30.CrossRefGoogle Scholar
Mazur, S., “Une remarque sur l'homeomorphie des champs fonctionels ”, Studia Math., 1 (1929), 83–85.CrossRefGoogle Scholar
Pugh, C., “On a theorem of P. Hartman”, Amer. J. Math., 91 (1969), 363–367.CrossRefGoogle Scholar
Riesz, F. and Sz.-Nagy, B., Functional analysis (Frederick Ungar, New York, 1955).Google Scholar
Stemberg, S., “Local contractions and a theorem of Poincaré ”, Amer. J. Math., 79 (1957), 809–824.CrossRefGoogle Scholar