On commutant of system of integration operators in multidimensional domains

Abstract

We describe the commutant of system of integration operators in the Frèchet space $H(\Omega)$ of all functions holomorphic in a domain $\Omega$ in $\mathbb C^N,$ which is polystar with respect to the origin. In particular, among such domains, there are the products of domains $\mathbb C$ being star with respect to the origin and complete Reinhardt domains with center at the origin. As in the one-dimensional case, the operators in the commutant are the Duhamel operators. We show that $H(\Omega)$ with the Duhamel product
$\ast$ is an associative and commutative topological algebra. It is topologically isomorphic to the commutant with the product, which is the composition of operators, and with the topology of bounded convergence. We obtain a similar to one--dimensional representation of the product $f\ast g$ as a sum containing one term being a multiple of $f$ and terms with the integrals at least in one variable of the function independent of the derivatives of $f$. By means of this representation we prove the criterion of $\ast$-invertibility of a function in $H(\Omega)$ and the corresponding convolution operator. We establish that the algebra $(H(\Omega), \ast)$ is local. In the case when the domain $\Omega$ is in addition convex, in the dual situation we obtain the criterion for the invertibility of operator from the commutant of system of operators of partial backward shift.