Duhamel Banach algebra structure of some space and related topics

Abstract

Let ? be a fixed complex number, and let ? be a simply connected region in complex plane C that is starlike with respect to ? ? ?. We define some Banach space of analytic functions on ? and prove that it is a Banach algebra with respect to the ?-Duhamel product defined by (f?? g)(z) := d/dz z?? f(z+??t)g(t)dt. We prove that its maximal ideal space consists of the homomorphism h? defined by h?(f)=f(?). Further, we characterize the lattice of invariant subspaces of the integration operator J?f(z)=?z? f(t)dt. Moreover, we describe in terms of ?-Duhamel operators the extended eigenvectors of J?.