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

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 $J_{A}P{J}_A^{\ast }$ 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 ${\bar{E}}^{\mathrm{\prime }}$, and of representable positive functionals on a ${}^{\ast }$-algebra.