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

TL;DR

This paper studies a class of operator-differential expressions involving bounded and finite-dimensional operators, providing regularization methods and proving the completeness of root functions for certain integral operators with specific boundary conditions.

Contribution

It introduces a new regularization approach for singular differential expressions with coefficients in negative Sobolev spaces and establishes the completeness of root functions for operators with Volterra integral components.

Findings

Regularization of singular differential expressions with negative Sobolev coefficients.

Proof of completeness of root functions for operators with Volterra integral operators.

Application to irregular semi-separated boundary conditions.

Abstract

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.