Spectrum of banach valued holomorphic functions and polar sets

Abstract

Let be a complex unital Banach algebra and D be a non-empty open connected subset of Let f be a holomorphic -valued function on D. We set: where is the set of all invertible elements in . In this paper, we give a description of using the classical spectrum of . Moreover, within the framework of this paper, we give a partial positive answer to the following problem which was posed by B. Aupetit: If the usual spectrum is polar for all is it true that is polar?