Coefficient multipliers with closed range
Abstract
For two power series
and
with positive radii of convergence, the Hadamard product or convolution is defined by
. We consider the prblem of characterizing those convolution operators
acting on spaces of holomorphic functions which have closed range. In particular, we show that every Euler differential operator
is a convolution operator
and we characterize the Euler differential operators, which are surjective on the space of holomorphic functions on every domain which contains the origin.
![f(z)=∑_{\nu=0}^∈fty f_\nu z<sup>\nu</sup>](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/678ed8b1b9be0a4cca8d56952c0c217c.png)
![g(z)=∑_{\nu=0}^∈fty g_\nu z<sup>\nu</sup>](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/82d4ecad62e1aef8c2e55726e09abba9.png)
![f\star g(z):=∑_{\nu=0}^∈fty f_\nu g_\nu z<sup>\nu</sup>](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/61837148cbba30bf7e9bc619accfc748.png)
![T<sub>f</sub>](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/2ebeb955d588ba1c64128e011a16607a.png)
![∑_{\nu=0}^∈fty \phi_\nu(z \frac{\partial}{\partial z})<sup>\nu</sup>](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/e9413bbe5eab5f45921ac7678e6ed700.png)
![T<sub>f</sub>](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/2ebeb955d588ba1c64128e011a16607a.png)
DOI Code:
10.1285/i15900932v17p61
Full Text: PDF