An analogue of the spectral mapping theorem for condition spectrum

Abstract

For 0 < ?? < 1, the ??-condition spectrum of an element a in a complex unital Banach algebra A is defined as, (Formula Presented). This is a generalization of the idea of spectrum introduced in [5]. This is expected to be useful in dealing with operator equations. In this paper we prove a mapping theorem for condition spectrum, extending an earlier result in [5]. Let f be an analytic function in an open set ?? containing ????(a). We study the relations between the sets ????(f(a)) and f(???? (a)). In general these two sets are different. We define functions ??(??), ?? (??) (that take small values for small values of ??) and prove that f(????(a)) ??? ????(??)(f(a)) and ????(f(a)) ??? f(?? ??(??)(a)). The classical Spectral Mapping Theorem is shown as a special case of this result. We give estimates for these functions in some special cases and finally illustrate the results by numerical computations.