Coefficient multipliers with closed range

Abstract

For two power series $f(z)=∑_{\nu=0}^∈fty f_\nu z^\nu$ and $g(z)=∑_{\nu=0}^∈fty g_\nu z^\nu$ with positive radii of convergence, the Hadamard product or convolution is defined by $f\star g(z):=∑_{\nu=0}^∈fty f_\nu g_\nu z^\nu$. We consider the prblem of characterizing those convolution operators $T_f$ acting on spaces of holomorphic functions which have closed range. In particular, we show that every Euler differential operator $∑_{\nu=0}^∈fty \phi_\nu(z \frac{\partial}{\partial z})^\nu$ is a convolution operator $T_f$ and we characterize the Euler differential operators, which are surjective on the space of holomorphic functions on every domain which contains the origin.