Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-26T00:42:17.993Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

[Barl] Barnes, B.A., Modular annihilator algebras, Canad. J. Math. 18(1966), 566–578.Google Scholar
[Bar2] , On the existence of minimal ideals in a Banach algebra, Trans. Amer. Math. Soc. 133(1968), 511–517.Google Scholar
[BD] Bonsall, F.F. and Duncan, J., Complete normed algebras, Springer-Verlag, Berlin, 1974.Google Scholar
[CD] Cohen, L.W. and Dunford, N., Transformations on sequence spaces, Duke Math. J. 3(1937), 689–701.Google Scholar
[GR] Giotopoulos, S. and Roumeliotis, M., Algebraic ideals ofsemiprime Banach algebras, Glasgow Math. J. 33(1991), 359–363.Google Scholar
[Jac] Jacobson, N., Structure of rings, Amer. Math. Soc. Colloq. Publ. 37, Providence, Rhode Island, 1956.Google Scholar
[KR] Katavolos, A. and Radjavi, H., Simultaneous triangularization of operators on a Banach space, J. London Math. Soc. (2) 41(1990), 547–554.Google Scholar
[Krup] Ya, N.. Krupnik, Banach algebras with symbol and singular integral operators, Birkhâuser Verlag, Basel, 1987.Google Scholar
[LNRR] Laurie, C., Nordgren, E., Radjavi, H. and Rosenthal, P., On triangularization of algebras of operators, J. Reine Angew. Math. 327(1981), 143–155.Google Scholar
[Mur] Murphy, G.J., Triangularizable algebras of compact operators, Proc. Amer. Math. Soc. (3) 84(1982), 354–356.Google Scholar
[Rie] Rickart, C.E., General theory of Banach algebras, D. Van Nostrand Company, Inc., Princeton, New Jersey, 1960.Google Scholar
[Rin] Ringrose, J.D., Compact non-self adjoint operators, D. Van Nostrand Company Inc., London, 1971.Google Scholar
[Row] Rowen, L.H., Polynomial identities in ring theory, Academic Press, Inc., New York, New York, 1980.Google Scholar