On the parallel sum of positive operators, forms, and functionals

TL;DR

This paper investigates the parallel sum of positive operators, forms, and functionals, providing factorizations that reveal their structural properties in Hilbert and Banach spaces.

Contribution

It introduces a new factorization approach for the parallel sum of positive operators, forms, and functionals, unifying various cases under a common framework.

Findings

Provides a factorization of the parallel sum as J_APJ_A^*

Extends the factorization to nonnegative Hermitian forms

Applies the results to positive functionals on *-algebras

Abstract

The parallel sum $A:B$ of two bounded positive linear operators $A,B$ on a Hilbert space $H$ is defined to be the positive operator having the quadratic form \begin{equation*} \inf\{(A(x-y)\,|\,x-y)+(By\,|\,y)\,|\,y\in H\} \end{equation*} for fixed $x\in H$. The purpose of this paper is to provide a factorization of the parallel sum of the form $J_APJ_A^*$ where $J_A$ is the embedding operator of an auxiliary Hilbert space associated with $A$ and $B$, and $P$ is an orthogonal projection onto a certain linear subspace of that Hilbert space. We give similar factorizations of the parallel sum of nonnegative Hermitian forms, positive operators of a complex Banach space $E$ into its topological anti-dual $\bar{E}'$, and of representable positive functionals on a $^*$-algebra.