The Krein-von Neumann Extension of a Regular Even Order Quasi-Differential Operator

TL;DR

This paper characterizes the Krein-von Neumann extension of a positive minimal operator associated with a regular even order quasi-differential expression, using boundary conditions and a special basis for the kernel.

Contribution

It provides a boundary condition-based characterization of the Krein-von Neumann extension for a class of quasi-differential operators, extending existing theory.

Findings

Boundary conditions for the Krein-von Neumann extension are explicitly described.

A basis for the kernel of the maximal operator is constructed.

The approach employs the Friedrichs extension description by Möller and Zettl.

Abstract

We characterize by boundary conditions the Krein-von Neumann extension of a strictly positive minimal operator corresponding to a regular even order quasi-differential expression of Shin-Zettl type. The characterization is stated in terms of a specially chosen basis for the kernel of the maximal operator and employs a description of the Friedrichs extension due to M\"oller and Zettl.