Hadamard type operators in spaces of holomorphic functions on a ball

Abstract

We study Hadamard type operators in the spaces of functions holomorphic in an open ball in $\mathbb{C}^N$ centered at the origin. These are continuous linear operators, for which each monomial is an eigenvector. We obtain a representation of Hadamard operators in the form of a multiplicative convolution. The proof of this representation employs essentially Fantappiè transformation realizing dual to the spaces of holomorphic functions and the holomorphy property of the characteristic function of a continuous linear operator in them. The applied method allows us to reduce the problem on representation of a Hadamard operator to the problem on holomorphic continuation of a function holomorphic at the point $0$ into a given open ball in $\mathbb{C}^N$ with $l_1$-norm. We prove that the space of the Hadamard type operators from one mentioned space into another with the topology of the bounded convergence is linearly topologically isomorphic to the strong dual to the space of the germs of all functions holomorphic on a closed polydisk.