On the Commutant of the Generalized Backward Shift Operator in Weighted Spaces of Entire Functions

TL;DR

This paper characterizes operators commuting with a generalized backward shift in weighted spaces of entire functions, providing conditions for isomorphisms, and explores their factorization and applications to the generalized Duhamel product.

Contribution

It offers necessary and sufficient conditions for commutant operators to be topological isomorphisms and investigates their factorization in weighted spaces of entire functions.

Findings

Operators with zeros in Q are classified into isomorphisms, surjective operators with finite-dimensional kernels, and finite-dimensional operators.

Conditions are established for when a commutant operator is a topological isomorphism.

Results are applied to analyze the generalized Duhamel product in holomorphic function spaces.

Abstract

We investigate continuous linear operators, which commute with the generalized backward shift operator (a one-dimensional perturbation of the Pommiez operator) in a countable inductive limit $E$ of weighted Banach spaces of entire functions. This space E is isomorphic with the help of the Fourier-Laplace transform to the strong dual of the Fr\'echet space of all holomorphic functions on a convex domain Q in the complex plane, containing the origin. Necessary and sufficient conditions are obtained that an operator of the mentioned commutant is a topological isomorphism of E. The problem of the factorization of nonzero operators of this commutant is investigated. In the case when the function, defining the generalized backward shift operator, has zeros in Q, they are divided into two classes: the first one consists of isomorphisms and surjective operators with a finite-dimensional kernel, and the second one contains finite-dimensional operators. Using obtained results, we study the generalized Duhamel product in Fr\'echet space of all holomorphic functions on Q.