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Some algebras of linear transformations which are not semi-simple 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 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.

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