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

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.