Theorems on polynomials in right invertible operators

Abstract

Suppose that $Q(D)$ is a polynomial in a right invertible operator acting in a linear space $X$, in general, with operator coefficients. Then $Q(D)=0$ if and only if $Q(D)R^{k}z=0$ for all $z\in \ker D(k=0,1,2,\dots )$ under appropriate assumptions on $X$, the right inverse $R$ of $D$ and coefficients of $Q(D)$.