Coefficient multipliers with closed range


Abstract


For two power series f(z)=∑_{\nu=0}^∈fty f_\nu z<sup>\nu</sup> and g(z)=∑_{\nu=0}^∈fty g_\nu z<sup>\nu</sup> 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<sup>\nu</sup>. We consider the prblem of characterizing those convolution operators T<sub>f</sub> 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})<sup>\nu</sup> is a convolution operator T<sub>f</sub> and we characterize the Euler differential operators, which are surjective on the space of holomorphic functions on every domain which contains the origin.

DOI Code: 10.1285/i15900932v17p61

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