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.