An analogue of the Conjecture of Dixmier is true for the algebra of polynomial integro-differential operators

Abstract

Let $A_1:=K\langle x, \frac{d}{dx} \rangle$ be the Weyl algebra and $\mI_1:= K\langle x, \frac{d}{dx}, \int \rangle$ be the algebra of polynomial integro-differential operators over a field $K$ of characteristic zero. The Conjecture/Problem of Dixmier (1968) [still open]: {\em is an algebra endomorphism of the Weyl algebra $A_1$ an automorphism?} The aim of the paper is to prove that {\em each algebra endomorphism of the algebra $\mI_1$ is an automorphism}. Notice that in contrast to the Weyl algebra $A_1$ the algebra $\mI_1$ is a non-simple, non-Noetherian algebra which is not a domain. Moreover, it contains infinite direct sums of nonzero left and right ideals.