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

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

$$\inf{(A(x - y) | (x - y) + ({By} | {y}) {y \in H}} $$

for fixed ${x \in H}$. The purpose of this paper is to provide a factorization of the parallel sum of the form ${J_A PJ_A^*}$ 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 ${\bar{E}^{\prime}}$, and of representable positive functionals on a ${^*}$-algebra.