In this thesis two problems concerning linear transformations are discussed. Both problems involve linear transformations which are not in some sense semi-simple; 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 semi-simple 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 non-semi- 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 quasi-nilpotent 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 semi-simple algebra generated by T and its associated projections as an algebra of functions. / Science, Faculty of / Mathematics, Department of / Graduate
| Identifer | oai:union.ndltd.org:UBC/oai:circle.library.ubc.ca:2429/40091 |
| Date | January 1958 |
| Creators | Langlands, Robert Phelan |
| Publisher | University of British Columbia |
| Source Sets | University of British Columbia |
| Language | English |
| Detected Language | English |
| Type | Text, Thesis/Dissertation |
| Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
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