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

We consider an operator-differential expression of the form $$\ell y=\frac{d^m}{dx^m}\Big(By^{(n)}+Cy\Big), \quad 0<x<1,$$ where $B$ is a linear bounded invertible operator, while $C$ is some finite-dimensional linear operator relatively bounded to the operator of $n$-fold differentiation. To such a form, we can reduce, in particular, various singular differential expressions with the coefficients in negative Sobolev spaces, 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 at the diagonal, we establish completeness of the root functions of an operator generated by the expression $\ell y$ and irregular semi-separated boundary conditions.