Finite sums of idempotents and logarithmic residues on connected domains

Abstract

We show that every finite sum of idempotents in a Banach algebra can be represented as the logarithmic residue of some analytic Banach algebra valued function defined on any, given bounded Cauchy domain. Moreover, using this, we can construct a non-invertible analytic Banach algebra valued function which is defined on any given bounded Cauchy domain and whose logarithmic residue is equal to zero., Consequently, the classical theorem concerning logarithmic residues fails in the general situation for all domains, in particular for connected domains.