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

Abstract

It is shown that in the spacesA _R (0 <R ⩽ ∞) 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$\varphi (D) = \sum\nolimits_{k = 0}^\infty \varphi _k D^k$ is equivalent to the operator Dⁿ (n is a fixed natural number) if and only if$\varphi (D) = \sum\nolimits_{k = 0}^n \varphi _k D^k$ and ¦ϕ _n ¦ = 1 for R < ∞ or ϕ _n ≠ 0 for R = ∞. Also the equivalence of two shift operators in the space A_∞ is investigated.