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Some algebras of linear transformations which are not semisimple Langlands, Robert Phelan
Abstract
In this thesis two problems concerning linear transformations are discussed. Both problems involve linear transformations which are not in some sense semisimple; otherwise they are unrelated. In part I we present a proof of the theorem that a linear transformation, of a finite dimensional vector space over a field, which has the property that the irreducible factors of its minimal polynomial are separable is the sum of a semisimple linear transformation and a nilpotent linear transformation, which commute with the original transformation and are polynomials in the original transformation.. We present an example to show that such a decomposition is not always possible. In parts II and III we obtain some representation theorems for closed algebras of linear transformations on a Banach space which are generated by spectral operators. Since such an algebra is the direct sum of its radical and a space of continuous functions its radical can be investigated more readily than the radical of an arbitrary nonsemi simple commutative Banach algebra. In part II we remark that the reduction theory for rings of operators allows one to reduce the problem of representing a spectral operator, T, on a Hilbert space to the problem of representing a quasinilpotent operator. When T is of type m+1 and has a "simple" spectrum it is quite easy to obtain an explicit representation of T. In part III we consider spectral operators on a Banach space. We impose quite stringent conditions, hoping that the theorems obtained for these special cases will serve as a model for more general theorems. The knowledge obtained at least delimits the possibilities. We assume that T is of type m+1 and has a "simple" spectrum. One other condition, which is satisfied if the space, X, on which T acts is separable, is imposed. We are then able to obtain a representation of X as a function space. These function spaces are modelled on the analogy of the Orlicz spaces. We are also able to obtain a representation of the not necessarily semisimple algebra generated by T and its associated projections as an algebra of functions.
Item Metadata
Title 
Some algebras of linear transformations which are not semisimple

Creator  
Publisher 
University of British Columbia

Date Issued 
1958

Description 
In this thesis two problems concerning linear transformations are discussed. Both problems involve linear transformations which are not in some sense semisimple; otherwise they are unrelated.
In part I we present a proof of the theorem that a linear transformation, of a finite dimensional vector space over a field, which has the property that the irreducible factors of its minimal polynomial are separable is the sum of a semisimple linear transformation and a nilpotent linear transformation, which commute with the original transformation and are polynomials in the original transformation.. We present an example to show that such a decomposition is not always possible.
In parts II and III we obtain some representation theorems for closed algebras of linear transformations on a Banach space which are generated by spectral operators. Since such an algebra is the direct sum of its radical and a space of continuous functions its radical can be investigated more readily than the radical of an arbitrary nonsemi simple commutative Banach algebra.
In part II we remark that the reduction theory for rings of operators allows one to reduce the problem of representing a spectral operator, T, on a Hilbert space to the problem of representing a quasinilpotent operator. When T is of type m+1 and has a "simple" spectrum it is quite easy to obtain an explicit representation of T. In part III we consider spectral operators on a Banach space. We impose quite stringent conditions, hoping that the theorems obtained for these special cases will serve as a model for more general theorems. The knowledge obtained at least delimits the possibilities. We assume that T is of type m+1 and has a "simple" spectrum. One other condition, which is satisfied if the space, X, on which T acts is separable, is imposed. We are then able to obtain a representation of X as a function space. These function spaces are modelled on the analogy of the Orlicz spaces. We are also able to obtain a representation of the not necessarily semisimple algebra generated by T and its associated projections as an algebra of functions.

Subject  
Genre  
Type  
Language 
eng

Date Available 
20120114

Provider 
Vancouver : University of British Columbia Library

Rights 
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.

DOI 
10.14288/1.0080637

URI  
Degree  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Campus  
Scholarly Level 
Graduate

Aggregated Source Repository 
DSpace

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Item Citations and Data
Rights
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.