Title page for ETD etd-08262010-161822


Type of Document Dissertation
Author Sutton, Daniel Joseph
URN etd-08262010-161822
Title Structure of Invariant Subspaces for Left-Invertible Operators on Hilbert Space
Degree PhD
Department Mathematics
Advisory Committee
Advisor Name Title
Ball, Joseph A. Committee Chair
Johnson, Martin E. Committee Member
Klaus, Martin Committee Member
Sun, Shu-Ming Committee Member
Keywords
  • Index
  • Wandering Subspace
  • Invariant Subspace
  • Weighted Shift
  • Left-Invertible
Date of Defense 2010-08-12
Availability unrestricted
Abstract
This dissertation is primarily concerned with studying the invariant subspaces of left-invertible, weighted shifts, with generalizations to left-invertible operators where applicable. The two main problems that are researched can be stated together as When does a weighted shift have the one-dimensional wandering subspace property for all of its closed, invariant subspaces? This can fail either by having a subspace that is not generated by its wandering subspace, or by having a subspace with an index greater than one. For the former we show that every left-invertible, weighted shift is similar to another weighted shift with a residual space, with respect to being generated by the wandering subspace, of dimension $n$, where $n$ is any finite number. For the latter we derive necessary and sufficient conditions for a pure, left-invertible operator with an index of one to have a closed, invariant subspace with an index greater than one. We use these conditions to show that if a closed, invariant subspace for an operator in a class of weighted shifts has a vector in $l^1$, then it must have an index equal to one, and to produce closed, invariant subspaces with an index of two for operators in another class of weighted shifts.

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