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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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