Theory Of Linear Transformations Research Articles

Spectral theory for operators on a Banach space

0. Introduction. The purpose of this paper is to study the spectral theory of a closed linear transformation T on a reflexive Banach space B. This will be done by means of certain vector-valued measures which are related to the transformation. (A set function m from the Borel sets of the complex plane to B will be called a vector-valued measure if the series EJ= j m (Si) converges to mr(Ui Si) for every sequence { Si I of disjoint Borel sets. The relevant properties of vector-valued measures are briefly derived in ?1(2). A vector-valued measure m will be called a T-measure if Tm(S) =fszdm(z) for all bounded Borel sets S. The properties of T-measures are studied in ?2. The results of ?2 are applied in ?3 to a class of transformations which have been called scalar-type transformations by Dunford [5], and which we call simply scalar transformations. A scalar operator as defined by Dunford is essentially one which admits a representation of the type t = fzdE(z), where E is a spectral measure. Unbounded scalar transformations have been studied by Taylor [16]. The main result of ?3 is Theorem 3.2, in which properties of the closures of certain sets of scalar transformations are derived. This theorem is actually a rather general spectral-type theorem, which has applications to several problems in the theory of linear transformations. As a corollary we obtain a well-known theorem, which might be called the spectral theorem for symmetric transformations, as given in Stone [15]. We also derive as a corollary the spectral theorem for self-adjoint transformations. Other corollaries of Theorem 3.2, which apply to results of Bade [2] and Halmos [9] are derived. In ?4 a functional calculus is developed for a class r of transformations T for which both T-measures and T*-measures exist in sufficient abundance. This is a very general functional calculus, so that correspondingly the usual theorems of functional calculus must be weakened if they are to remain true. There is a generalization of the concept of a T-measure introduced in ?5. Many theorems proved in ?2 have analogues which hold after the generalization. The new type of vector-valued measures have much the same relation

The spectrum of linear transformations

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

Linear Transformations and Geometry

The following note suggests certain connections between the theory of linear transformations and quadratic forms on the one hand, and the geometry of second degree surfaces on the other. It is hoped that the note may prove useful to those who may have to teach either theory to students who already possess an elementary knowledge of the other. The general ideas may be such as may well have occurred to anyone familiar with both theories, but the examples given may be new to readers.