Ph.D. (Doctor of Philosophy)
Department of Mathematical Sciences
Banach spaces||Linear operators||Invariant subspaces
Let X be a complex, infinite-dimensional Banach space and T : X → X a continuous linear operator. A closed subspace Y of X such that Y and X/Y both have infinite dimension is called a halfspace. We say that Y is almost-invariant under T if there is a finite-rank linear operator F : X → X such that (T + F)Y ⊆ Y; in this case, the minimum possible rank of F is called the defect. If (T +K)Y ⊆ Y for some compact linear operator K : X → X then we say that Y is essentially-invariant under T. We show that if T is compact, strictly singular, weakly compact, quasinilpotent, or has at most countably many eigenvalues, then it admits an almost-invariant halfspace (AIHS) of defect [precedes or equal to] 1. We also improve an existing result by showing that T* always admits an AIHS of defect [precedes or equal to] 1, with no assumptions on T. Although it is not currently known whether every operator acting on a complex, infinite-dimensional Banach space admits an AIHS, we do nevertheless show that every such operator admits an essentially-invariant halfspace. Some additional results on AIHS's and their connection to invariant subspaces are also obtained.
Wallis, Ben, "The almost-invariant halfspace problem" (2016). Graduate Research Theses & Dissertations. 5297.
Northern Illinois University
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