Duhamel Banach algebra structure of some space and related topics

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