Compound operator equations in generalized derivatives and their applications to Appell sequences

Yu. F. Korobeinik

Sbornik. Mathematics, Tome 31 (1977) no. 4, pp. 425-443

Résumé

Let $E$ be a vector space of sequences of numbers, containing all of the basis vectors $e_k$, with the Köthe topology $\nu$; let $\{f_k\}$ be a fixed sequence of nonzero complex numbers; let $D$ be a Gel'fond–Leont'ev generalized differentiation operator: $$ (Dc)_k=\frac{f_k}{f_{k+1}}c_{k+1},\qquad k=0,1,2,\dots, $$ and let $p$ be an operator of the form $(p_c)_m=(-1)^m, m=0,1,\dots$ . In this work there is an investigation of an infinite-order operator $$ Lc=\sum_{k=0}^\infty a_kD^kc+\sum_{k=0}^\infty b_kD^kP_c. $$ Under rather general assumptions it is shown that $L_0$ is an epimorphism of $(E,\nu)$, and the kernel is described; conditions are established for $L_0$ to be an isomorphism of $(E,\nu)$. On the basis of these results criteria are found for an Appell sequence to be a quasi-power basis or representing system in $(E,\nu)$. Bibliography: 16 titles.