Operator-differential expressions: regularization and completeness of the root functions

We consider an operator-differential expression of the form $$ \ell y=\frac{d^m}{dx^m}(By^{(n)}+Cy), \qquad 0<x<1, $$ where $B$ is a linear bounded invertible operator, while $C$ is some finite-dimensional linear operator relatively bounded with respect to the operator of $n$-fold differentiation. We establish that, in particular, various singular differential expressions with coefficients in negative Sobolev spaces can be reduced to this form, which creates an alternative to their regularization. In the case when $B$ is an integral Volterra operator of the second kind with a continuous kernel vanishing identically on the diagonal, we prove the completeness of the system of eigen- and associated functions of the operator generated by the expression $\ell y$ and irregular semi-separated boundary conditions. Bibliography: 94 titles.