On the equivalence of differential operators of infinite order in analytic spaces

It is shown that in the spaces $A_R$ ($0<R\le\infty$) of all functions which are single-valued and analytic in the disk $|z|<R$ with the topology of compact convergence, the differential operator of infinite order with constant coefficients $\varphi(D)=\sum_{k=0}^\infty\varphi_kD^k$ is equivalent to the operator $D^n$ ($n$ is a fixed natural number) if and only if $\varphi(D)=\sum_{k=0}^n\varphi_kD^k$ and $|\varphi_n|=1$ for $R<\infty$ or $\varphi\ne0$ for $R=\infty$. Also the equivalence of two shift operators in the space $A_\infty$ is investigated.