Regularity of Generalized Scalar Operators with Spectrum Contained in a Line
نویسنده
چکیده
Generalized scalar operators were introduced by C. Foia~ [4] and a detailed study of such operators can be found in the monograph [3]. An important subclass consists of regular generalized scalar operators which enjoy properties not shared by all generalized scalar operators. For example, the sum and product of commuting generalized scalar operators S and T need not be generalized scalar operators[2; §3]. However, if, in addition, S and T are both regular, then ST and S + T are again generalized scalar operators [3; p.l06], although they need not be regular [2; §3]. In particular, there exist generalized scalar operators which are not regular [1,2]. A closed subset F of the complex plane ~ is called thin [3; p.lOO] if the function A + ~ on F ( the bar denotes complex conjugation ) is the restriction of a function which is analytic in a neighbourhood of F. It is clear that any closed subset of a thin set is also a thin set and that segments of a line are thin sets. Accordingly, the following result is an immediate consequence of Theorem 4.1.11 in [3].
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