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Invariant Complements to Closed Invariant Subspaces

Published online by Cambridge University Press:  20 November 2018

M. P. Thomas
M. P. Thomas*
Affiliation:
The University of Texas at Austin, Austin, Texas
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The question under what conditions a closed invariant subspace possesses a closed invariant complement is of major importance in operator theory. In general it remains unanswered. In this paper we drop the requirement that the invariant complement be closed. We show in section 1 that the question is answerable under fairly mild conditions for a quasinilpotent operator (Theorem 1.5). These conditions will cover the case of a quasinilpotent operator with dense range and no point spectrum. In section 2 we discuss the consequences for the Volterra operator V. Since V is unicellular, its proper closed invariant subspaces do not possess closed invariant complements. However, they are all algebraically complemented (Proposition 2.1).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 31 , Issue 1 , 01 February 1979 , pp. 139 - 147
Copyright
Copyright © Canadian Mathematical Society 1979

References

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