tailieunhanh - Báo cáo toán học: "Extended spectral operators "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Mở rộng phổ các nhà khai thác. | J. OPERATOR THEORY 9 1983 269 - 296 Copyright by INCREST 1983 EXTENDED SPECTRAL OPERATORS w. RICKER 0. INTRODUCTION The spectral theory of non-selfadjoint operators has been developed by N. Dunford and others. A comprehensive treatment can be found in the monographs of Dunford and Schwartz 4 The central concept is that of a spectral operator. For a spectral operator there exists a decomposition of the space into a discrete or continuous direct sum of reducing subspaces in each of which the operator acts in a particularly simple way. This decomposition of the space is realised by a projection valued measure called the resolution of the identity for that operator. However as pointed out by N. Dunford in the survey 2 many interesting and important operators of analysis may fail to be spectral in this sense. For this reason it is desirable to have a more general concept of spectral operator. This note is devoted to extending the notion of spectral operator in the sense of Dunford. For an operator T in a space X we shall seek a space Y containing a copy of X such that T has a natural extension to a spectral operator on Y. That is the space X is continuously included in a space Y which accomodates the resolution of the identity for T. For example let T be the operator on the space X L 0 l given by T f g where g t t e 0 1 . A prospective resolution of the identity for T is defined by P E f f eX for each Borel subset E of 0 1 . However the map E P E is not ơ-additive and so T is not a spectral operator in X. All the projections needed to form the resolution of the identity are available but the topology of X is too strong to allow p to be countably additive. This can be simply remedied by equipping X with a topology weaker than its norm topology. Many differential operators T arising from boundary value problems are almost spectral operators see Example below . They have associated with them a large family of projection operators which are in a certain sense dense in