The spectrum of linear transformations

Abstract

The determination of the structure of a linear transformation T which maps a complex vector space Q3 into itself is naturally a primary object of the theory of linear transformations. The first stage of such a study centers about the question of reducibility. The concept may be introduced in various ways. In one formulation it is required to find all projections (bounded idempotent transformations) P which commute with T. At a more incisive level, one seeks those pairs of manifolds 9) and 1 (not necessarily disjoint) which split the space and are transformed into themselves by T. For transformations in general vector spaces the known results all refer to special types such as the completely continuous or the weakly almost periodic. This paper will deal almost exclusively with the first type of reducibility of the general bounded transformation. The boundedness of T is assumed for convenience; in case merely its closure is hypothesized the salient features of the theory are still valid. The results are all based on one method, that of a contour integral of the resolvent of T. They seem to exhaust the possibilities for this particular tool. Means for cracking the spectrum directly would undoubtedly have to be of a much more delicate nature. The fundamental projection is the integral