On the equivalence in \(A_ r\) of an integro-differential operator of a certain kind and the Euler operator (Q791820)

In the last twenty years or so, quite a few research papers deal with studies of various properties of differential and integro-differential operators as well as with the equivalence between some of them [See, for instance, \textit{I. I. Ibragimov} and \textit{I. F. Kushnirchuk}, Dokl. Akad. Nauk SSSR 214, 33-36 (1974; Zbl 0312.47039); \textit{I. F. Kushnirchuk} and \textit{K. M. Fishman}, Izv. Vyssh. Uchebn. Zaved., Mat. 1976, No.8(171), 42-51 (1976; Zbl 0352.46013) and \textit{I. F. Kushnirchuk}, Sib. Mat. Zh. 18, 340-347 (1977; Zbl 0362.34027)]. The author deals with the Volterra type integro-differential operator with a singularity \[ \ell [y]=z^{\tilde n}D^ n_ zy+\sum^{n}_{m=1}z^{n-m}Q_ m(z)D_ z^{n-m}y+\int^{z}_{0}N(z,\xi)y(\xi)d\xi \] and the Euler operator \[ L_ 0=z^ nD^ n+z^{n-1}Q_ 1(0)D_ z^{n-1}+...+zQ_{n- 1}(0)D_ z+Q_ n(0)E, \] where \(D_ z=d/dz\), E is the identity, \(n\geq 2\), \(Q_ m(z)\in A_ R (m=1,2,..,n)\), N(z,\(\xi)\) is an analytic function in the bicylinder \(\{| z|<R,| \xi |<R\}\). If \(\alpha_ k=A_ k^ n+\sum^{n}_{m=1}Q_ m(0)A_ k^{n-m}\), \(A_ k^{\delta}=k(k-1)...(k-s+1)\), \(k\geq s\), \(A_ k^ 0=1\), \(Q_ m(z)\) does satisfy the condition \(\alpha_ q\neq \alpha_ k\), \(q\neq k,q,k=0,1,2,..\). (The values \(Q_ 1(0)=0,-1,-2,...\), are excluded for \(n=2\), because in this case the condition \(\alpha_ q\neq \alpha_ k,q\neq k,k=0,1,2,..\). is not satisfied for some q and k). The main theorem of this paper asserts that the operators \(\ell\) and \(L_ 0\) are equivalent in \(A_ R\), where \(A_ R\) is the space with compact topology of all analytic functions in the unit circle \(| z|<R\), \(0<R<\infty\).