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Order of singularity applied to linear operators

Published online by Cambridge University Press:  14 November 2011

James E. Scroggs
James E. Scroggs
Affiliation:
University of Arkansas, Fayetteville, Arkansas 72701, U.S.A.
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Synopsis

We extend the notion of order of singularity for complex-valued functions, as originally set forth by Hadamard, to Banach algebra-valued functions. Restricting our attention to linear operators whose spectral radius is one, we obtain a connection between the rate of growth of the norm of iterates of a linear operator and the rate of growth of the norm of the resolvent of the operator near the spectrum of the operator. In the finite dimensional case, we obtain an upper bound on the size of the Jordan block corresponding to an eigenvalue of maximum modulus.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 106 , Issue 1-2 , 1987 , pp. 103 - 111
Copyright
Copyright © Royal Society of Edinburgh 1987

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References

1Bartle, Robert G.. Spectral localization of operators in Banach spaces. Math. Ann. 153 (1964), 261–269.CrossRefGoogle Scholar
2Dunford, Nelson and Schwartz, J. T.. Linear Operators, Part III (New York: Wiley Interscience, 1971).Google Scholar
3Hadamard, Jaques. Essai sur l'etude des fonctions donnees par leur developpement de Taylor, Oeuvres de Jacques Hadamard, pp. 7–92 (Paris: Centre National de la Recherche Scientifique, 1968).Google Scholar
4Leaf, G. K.. A spectral theory for a class of linear operators. Pacific J. Math. 13 (1963), 141–155.CrossRefGoogle Scholar
5Leaf, G. K.. An approximation theorem for a class of operators. Proc. Amer. Math. Soc. 16 (1965), 991–995.CrossRefGoogle Scholar
6Mandelbrojt, Szolem. Modern Researches on Singularities of Functions Defined by Taylor Series. The Rice Institute Pamphlet, 14, No. 4 (1927), 290–312.Google Scholar
7Stampfli, J. G.. A local spectral theory for operators III; resolvents, spectral sets and similarity. Trans. Amer. Math. Soc. 168 (1972), 133–151.Google Scholar
8Taylor, A. E.. Introduction to Functional Analysis (New York: John Wiley, 1966).Google Scholar
9Wermer, J.. The existence of invariant subspaces. Duke Math. J. 19 (1952), 615–622.CrossRefGoogle Scholar
10Wolf, F.. Operators in Banach spaces that admit a generalized decomposition. Indag. Math. 19 (1957), 302–311.CrossRefGoogle Scholar