Shorting, parallel addition and form sums of nonnegative selfadjoint linear relations

TL;DR

This paper extends the concepts of shorting and parallel addition to nonnegative selfadjoint linear relations, exploring their properties, connections with Cayley transforms, and applications to selfadjoint extensions.

Contribution

It introduces new operations and properties for nonnegative selfadjoint linear relations, expanding the theoretical framework beyond bounded operators.

Findings

Existence of two arithmetic-harmonic means for pairs of relations

Connections established between these operations and Cayley transforms

Applications to nonnegative selfadjoint extensions of symmetric relations

Abstract

We extend the operations of shorting and parallel addition from the cone of bounded nonnegative selfadjoint operators in a Hilbert space to the set of all nonnegative selfadjoint linear relations. New properties of these operations and connections with the Cayley transforms are established. It is shown that for a pair of nonnegative selfadjoint linear relations there exist, in general, two arithmetic--harmonic means. Applications of the arithmetic, harmonic, and arithmetic--harmonic means to the theory of nonnegative selfadjoint extensions of nonnegative symmetric linear relations are given.