Geodesic
On the equivalence of differential operators of infinite order in analytic spaces
Matematičeskie zametki, Tome 21 (1977) no. 1, pp. 33-39 
N. I. Nagnibida; N. P. Oliinyk. On the equivalence of differential operators of infinite order in analytic spaces. Matematičeskie zametki, Tome 21 (1977) no. 1, pp. 33-39. http://geodesic.mathdoc.fr/item/MZM_1977_21_1_a3/
@article{MZM_1977_21_1_a3,
     author = {N. I. Nagnibida and N. P. Oliinyk},
     title = {On the equivalence of differential operators of infinite order in analytic spaces},
     journal = {Matemati\v{c}eskie zametki},
     pages = {33--39},
     year = {1977},
     volume = {21},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1977_21_1_a3/}
}
TY  - JOUR
AU  - N. I. Nagnibida
AU  - N. P. Oliinyk
TI  - On the equivalence of differential operators of infinite order in analytic spaces
JO  - Matematičeskie zametki
PY  - 1977
SP  - 33
EP  - 39
VL  - 21
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/MZM_1977_21_1_a3/
LA  - ru
ID  - MZM_1977_21_1_a3
ER  - 
%0 Journal Article
%A N. I. Nagnibida
%A N. P. Oliinyk
%T On the equivalence of differential operators of infinite order in analytic spaces
%J Matematičeskie zametki
%D 1977
%P 33-39
%V 21
%N 1
%U http://geodesic.mathdoc.fr/item/MZM_1977_21_1_a3/
%G ru
%F MZM_1977_21_1_a3

Voir la notice de l'article provenant de la source Math-Net.Ru

It is shown that in the spaces $A_R$ ($0) of all functions which are single-valued and analytic in the disk $|z| 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.