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

Tarcsay, Zs.

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 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.

OPERATOR theory; FUNCTIONALS; HILBERT space; FACTORIZATION; ORTHOGRAPHIC projection

Acta Mathematica Hungarica, 2015, Vol 147, Issue 2, p408

0236-5294

Academic Journal

10.1007/s10474-015-0533-6