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