The Krein transform and semi-bounded extensions of semi-bounded linear relations

TL;DR

This paper explores the Krein transform's role in establishing a correspondence between positive relations and symmetric contractions, and develops a theory for semi-bounded extensions of semi-bounded linear relations.

Contribution

It introduces a variation of the Krein transform that creates an involution for linear relations and formulates a theory for semi-bounded symmetric extensions.

Findings

Provides a formula for positive extensions of quasi-null relations

Establishes a one-to-one correspondence via the Krein transform

Develops a theory for semi-bounded symmetric extensions

Abstract

The Krein transform is the real counterpart of the Cayley transform and gives a one-to-one correspondence between the positive relations and symmetric contractions. It is treated with a slight variation of the usual one, resulting in an involution for linear relations. On the other hand, a semi-bounded linear relation has closed semi-bounded symmetric extensions with semi-bounded selfadjoint extensions. A self-consistent theory of semi-bounded symmetric extensions of semi-bounded linear relations is presented. By using The Krein transform, a formula of positive extensions of quasi-null relations is provided.