Regular Ordinary Differential Operators with Involution

Abstract

The main results of the paper are related to the study of differential operators of the form

\(Ly = {y^{\left( n \right)}}\left( { - x} \right) + \sum\limits_{k = 1}^n {pk\left( x \right){y^{\left( {n - k} \right)}}\left( { - x} \right) + } \sum\limits_{k = 1}^n {{q_k}\left( x \right){y^{\left( {n - k} \right)}}} \left( x \right),\,x \in \left[ { - 1,1} \right],\)

with boundary conditions of general form concentrated at the endpoints of a closed interval. Two equivalent definitions of the regularity of boundary conditions for the operator L are given, and a theorem on the unconditional basis property with brackets of the generalized eigenfunctions of the operator L in the case of regular boundary conditions is proved.