An Analogue of the Spectral Mapping Theorem for Condition Spectrum

Abstract

For 0 < ϵ < 1, the ϵ-condition spectrum of an element a in a complex unital Banach algebra A is defined as, \(\sigma_{\epsilon}(a)=\{\lambda\,\epsilon\,\mathbb{C}:\lambda-\hbox {a is not invertible or} \parallel\lambda-a\parallel\parallel(\lambda-a)^{-1}\parallel\geq\frac{1}{\epsilon}\}.\) 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 9 containing σϵ(a). We study the relations between the sets \(\sigma_{\epsilon}(\tilde{f}(a))\,\,\, and \,\,\,f((\sigma)_{\epsilon}(a)).\) In general these two sets are different. We define functions \(\phi(\epsilon), \psi(\epsilon)\) (that take small values for small values of ϵ) and prove that \( f (\sigma_{\epsilon}(a))\subseteq \sigma_{\phi}(\epsilon)(\tilde{f}(a))\; \rm and \;\sigma_{\epsilon}(\tilde{f}(a))\subseteq f(\sigma_{\psi}(\epsilon)(a)).\) 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.KeywordsSpectrumanalytic functioncondition spectrumSpectral Mapping Theorem