Citation
Taylor, Richard Forsythe (1968) Invarient subspaces in Hilbert and normed spaces. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd10042002144336
Abstract
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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 [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 quasinilpotent 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 
Major Option:  Mathematics 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  11 March 1968 
Record Number:  CaltechETD:etd10042002144336 
Persistent URL:  http://resolver.caltech.edu/CaltechETD:etd10042002144336 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  3899 
Collection:  CaltechTHESIS 
Deposited By:  Imported from ETDdb 
Deposited On:  07 Oct 2002 
Last Modified:  26 Dec 2012 03:03 
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