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Invariant subspaces of certain classes of operators Open Access


Other title
Invariant subspaces
Nonnegative matrices
Pure mathematics
Banach spaces
Positive operators
Matrix semigroups
Strictly singular operators
Operator theory
Functional Analysis
Almost invariant subspaces
Type of item
Degree grantor
University of Alberta
Author or creator
Popov, Alexey
Supervisor and department
Troitsky, Vladimir G. (Mathematical and Statistical Sciences)
Examining committee member and department
Tomczak-Jaegermann, Nicole (Mathematical and Statistical Sciences)
Pogosyan, Dmitri (Physics)
Dilworth, Stephen (Department of Mathematics, University of South Carolina)
Litvak, Alexander (Mathematical and Statistical Sciences)
Runde, Volker (Mathematical and Statistical Sciences)
Department of Mathematical and Statistical Sciences

Date accepted
Graduation date
Doctor of Philosophy
Degree level
The first part of the thesis studies invariant subspaces of strictly singular operators. By a celebrated result of Aronszajn and Smith, every compact operator has an invariant subspace. There are two classes of operators which are close to compact operators: strictly singular and finitely strictly singular operators. Pelczynski asked whether every strictly singular operator has an invariant subspace. This question was answered by Read in the negative. We answer the same question for finitely strictly singular operators, also in the negative. We also study Schreier singular operators. We show that this subclass of strictly singular operators is closed under multiplication by bounded operators. In addition, we find some sufficient conditions for a product of Schreier singular operators to be compact. The second part studies almost invariant subspaces. A subspace Y of a Banach space is almost invariant under an operator T if TY is a subspace of Y+F for some finite-dimensional subspace F ("error"). Almost invariant subspaces of weighted shift operators are investigated. We also study almost invariant subspaces of algebras of operators. We establish that if an algebra is norm closed then the dimensions of "errors" for the operators in the algebra are uniformly bounded. We obtain that under certain conditions, if an algebra of operators has an almost invariant subspace then it also has an invariant subspace. Also, we study the question of whether an algebra and its closure have the same almost invariant subspaces. The last two parts study collections of positive operators (including positive matrices) and their invariant subspaces. A version of Lomonosov theorem about dual algebras is obtained for collections of positive operators. Properties of indecomposable (i.e., having no common invariant order ideals) semigroups of nonnegative matrices are studied. It is shown that the "smallness" (in various senses) of some entries of matrices in an indecomposable semigroup of positive matrices implies the "smallness" of the entire semigroup.
Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. Where the thesis is converted to, or otherwise made available in digital form, the University of Alberta will advise potential users of the thesis of these terms. The author reserves all other publication and other rights in association with the copyright in the thesis and, except as herein before provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author's prior written permission.
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