On positive decompositions and proper splittings of Hermitian operators

TL;DR

This paper explores positive decompositions and proper splittings of Hermitian operators, providing new characterizations and conditions for convergence, which enhance understanding of operator positivity and decomposition methods in Hilbert spaces.

Contribution

It introduces a norm-based characterization of Hermitian positivity and links positive orthogonal decompositions to proper splittings, including convergence criteria.

Findings

Positivity characterized via a norm condition involving pseudo polar decomposition.

Relation established between positive orthogonal decompositions and proper splittings.

Sufficient condition for convergence of proper splittings in bounded linear operators.

Abstract

In this article we study different aspects of Hermitian operators applying the concept of positive decompositions. On the one hand, we characterize the positivity of an Hermitian operator by means of a norm condition where the factors of certain pseudo polar decomposition of the operator, are involved. On the other hand, we relate the concept of positive orthogonal decomposition of Hermitian operators to the notion of proper splittings of operators. Furthermore, we present a sufficient condition for the convergence of proper splittings for general bounded linear operators on Hilbert spaces.