On positive decompositions and proper splittings of Hermitian operators

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.