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Invariant subspaces of certain classes of operators

  • Author / Creator
    Popov, Alexey
  • 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.

  • Subjects / Keywords
  • Graduation date
    2011-06
  • Type of Item
    Thesis
  • Degree
    Doctor of Philosophy
  • DOI
    https://doi.org/10.7939/R3ND5S
  • License
    This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for non-commercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.
  • Language
    English
  • Institution
    University of Alberta
  • Degree level
    Doctoral
  • Department
    • Department of Mathematical and Statistical Sciences
  • Supervisor / co-supervisor and their department(s)
    • Troitsky, Vladimir G. (Mathematical and Statistical Sciences)
  • Examining committee members and their departments
    • Tomczak-Jaegermann, Nicole (Mathematical and Statistical Sciences)
    • Runde, Volker (Mathematical and Statistical Sciences)
    • Pogosyan, Dmitri (Physics)
    • Litvak, Alexander (Mathematical and Statistical Sciences)
    • Dilworth, Stephen (Department of Mathematics, University of South Carolina)