On roots of the multiple integration operator in the space of functions analytic in a disk

Let $A_R$ denote the space of all single-valued functions analytic in the disk $|z|<R$, $0<R\leqslant\infty$, with the topology of compact convergence, and let $J$, $J\cdot=\int_0^z\cdot\,d\xi$, be the integration operator on it. In the paper all continuous linear operators on $A_R$ which satisfy the condition $Y^p=J^p$, where $p$ is a fixed natural number, are found, and it is shown that for each of them there exists a one-to-one bicontinuous mapping $T$ of the space $A_R$ to itself which commutes with $J^p$ and satisfies $YT=TJ$.
Bibliography: 8 titles.