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Elementary Chains of Invariant Subspaces of a Banach Space

Published online by Cambridge University Press:  20 November 2018

Jon M. Clauss
Jon M. Clauss*
Affiliation:
Department of Mathematics University of Oregon Eugene, Oregon 97403-1222 U.S.A. e-mail: clauss@euclid. uoregon. edu Mathematics and Computer Science Department Augustana College Rock Island, Illinois 61201 U.S.A. e–mail: maclauss@augustana.edu
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Abstract

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We will generalize Ringrose's notion of a simple chain of closed invariant subspaces of a compact operator acting on a Banach space, to that of an elementary chain of invariant subspaces of a subalgebra of compact operators. With this we expand the notion of diagonal coefficients to that of diagonal representations and subsequently generalize Ringrose's theorem equating the spectrum of an operator to the collection of diagonal coefficients. This in turn, in conjunction with some results from the theory of Polynomial Identity algebras, allows us to generalize Murphy's theorem which states that a closed subalgebra of compact operators is simultaneously triangularizable if and only if / rad () is commutative. Let be an algebra of compact operators acting on a Banach space with a norm ‖ · ‖ which dominates the operator norm, and under which 𝒜 is complete. Then has an elementary chain of invariant subspaces of bound n if and only if / rad () satisfies the standard polynomial .

Keywords

47A15
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 47 , Issue 2 , 01 April 1995 , pp. 290 - 301
Copyright
Copyright © Canadian Mathematical Society 1995

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