Taylor, Richard Forsythe (1968) Invarient subspaces in Hilbert and normed spaces. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd-10042002-144336
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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  and Bernstein and Robinson . The entire thesis is exposited within the framework of nonstandard analysis as developed by Robinson .
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  and has been published in preprint form by Gillespie , 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.
|Item Type:||Thesis (Dissertation (Ph.D.))|
|Degree Grantor:||California Institute of Technology|
|Division:||Physics, Mathematics and Astronomy|
|Thesis Availability:||Public (worldwide access)|
|Defense Date:||11 March 1968|
|Default Usage Policy:||No commercial reproduction, distribution, display or performance rights in this work are provided.|
|Deposited By:||Imported from ETD-db|
|Deposited On:||07 Oct 2002|
|Last Modified:||26 Dec 2012 03:03|
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