Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Các nhà khai thác với kỳ dị quang phổ. | J. OPERATOR THEORY 15 1986 307-325 Copyright by INCREST 1986 OPERATORS WITH SPECTRAL SINGULARITIES B. NAGY 1. INTRODUCTION It is the basic idea in the theory of spectral operators that for a large number of linear operators the closed spectral operators in a complex Banach space in the sense of Dunford and Bade cf. 5 there exists a strongly countably additive spectral measure defined on the ơ-algebra of all Borel subsets of the complex plane which is closely connected with the given operator. J. T. Schwartz 17 noticed first that for some operators even in Hilbert space the construction of such a completely satisfying spectral measure is impeded by the fact that the natural candidate for the corresponding measure becomes unbounded in the neighborhood of and cannot be defined for some Borel subset of the spectrum. Similar phenomena have been pointed out by Ljance 11 in connection with a class of second-order non-selfadjoint ordinary differential operators and by several authors cf. Harvey 8 Langer 10 studying definitizable operators in Krein spaces. Attempting to construct a general theory for such bounded operators Bacalu 1 2 and the present author 12 have studied the so called S-spectral operators for which the spectral measure is defined only for a smaller ơ-algebra of the Borel subsets which avoid or contain a bad exceptional closed subset s of the spectrum. However such spectral measures must still be uniformly bounded so cannot play the role of the measures mentioned in the preceding paragraph. The main idea in this paper is that we consider such spectral measures more exactly resolutions of the identity that are defined and strongly countably additive only on a Boolean algebra of the Borel subsets of the complex plane which in some specified sense avoid or contain a bad exceptional closed subset S . Such measures can already be not uniformly bounded moreover it will be shown that for each closed operator there is a smallest one among these bad closed sets. It
