Characterization of continuous endomorphisms in the space of entire functions of a given order

TL;DR

This paper characterizes continuous endomorphisms in the space of entire functions of a given exponential type and order, showing they correspond to specific infinite-order differential operators with controlled growth.

Contribution

It establishes a one-to-one correspondence between continuous endomorphisms and certain infinite-order differential operators with growth conditions.

Findings

Unique differential operator representation of endomorphisms

Growth conditions on coefficients ensure continuity

Conversely, such operators induce continuous endomorphisms

Abstract

The aim of this paper is to characterize continuous endomorphisms in the space of entire functions of exponential type of order $p>0$. Let $A_p$ denote the space of entire functions of $n$ complex variables $z\in{\mathbb C}^n$ of order $p$ of normal type. We consider an endomorphism $F$ in the space, which is considered to be a DFS-space. We show that there is a unique linear differential operator $P$ of infinite order with coefficients in the space which realizes $F$, that is, $Ff=Pf$ holds for any $f\in A_p$. The coefficients satisfy certain growth conditions and conversely, if a formal differential operator of infinite order with coefficients in $A_p$ satisfy these conditions, then it induces a continuous endomorphism.