The problem of representing an arbitrary linear operator in the form of a differential operator of infinite order

We consider linear operators, acting continuously from the space $A_{R_1}$ of functions analytic in the disk $|z|<R_1$ into the space $A_{R_2}$. We show that every such operator may be represented in the form of a linear differential operator of infinite order with coefficients analytic in the disk $|z|<R_2$.