Invariant Subspaces in Hilbert and Normed Spaces

Citation

Taylor, Richard Forsythe (1968) Invariant Subspaces in Hilbert and Normed Spaces. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/1453-JV44. https://resolver.caltech.edu/CaltechETD:etd-10042002-144336

Abstract

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. This dissertation concerns itself with the following question: Suppose T is a bounded linear operator from an infinite dimensional Hilbert Space into itself. What are sufficient conditions to imply the existence of a nonzero, proper subspace M of H such that T(M)[...]M? The methodology used to approach the question is in line with the methods developed by Aronzajn and Smith [1] and Bernstein and Robinson [3]. The entire thesis is exposited within the framework of nonstandard analysis as developed by Robinson [9]. Chapter 1 of the dissertation develops the necessary theory involved, and presents a necessary and sufficient condition for T to have a proper invariant subspace. The conditions involve assumptions on certain finite dimensional approximations of T. Chapter 2 demonstrates two situations under which the conditions presented in Chapter 1 come about. The first of these, which was announced by Feldman [5] and has been published in preprint form by Gillespie [6], was proved independently by the author under more relaxed conditions. For simplicity, we state here the Feldman result. Theorem: If T is quasi-nilpotent and if the algebra generated by T has a nonzero compact operator in its uniform closure, then T has an invariant subspace. It is still an open question whether or not the condition "T commutes with a compact operator" implies the desired result. By insisting that C be "very compact" (to be defined) the following result is demonstrated. Theorem: If C is a nonzero "very compact" operator, and if TC=CT, then T has an invariant subspace.

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