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?.
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