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

الأصل · EN

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 equation* {(A(x-y)|x-y)+(By|y)|y∈ H} equation* for fixed x∈ H. The purpose of this paper is to provide a factorization of the parallel sum of the form JₐPJₐ* where Jₐ 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 E', and of representable positive functionals on a *-algebra.