Extensions of the Fill–Fishkind formula and the infimum – parallel sum relation

Operators with compatible ranges

“…The second reason for such generalization comes from the results of [13,11,14,12] which describe different properties of operators A and B which coincide on R(A * ) ∩ R(B * ) (a generalization of Werener's condition of weak complementarity, see [20]). Accordingly, we will present different properties of CoR operators, regarding range additivity, some additive results for the Moore-Penrose inverse, etc.…”

“…(1) P AP = AP = A * P = P A * P . If R(T ) is closed, then also P ∆P = ∆P ; In order to give a formula for (T +T * ) † when T is CoR, we first prove the following result regarding range additivity, explaining when does (T + T * ) † exist (see also [11,Theorem 2.4]). Proof.…”

“…Formula ( . In fact, the last expression gives the parallel sum of T and T * as defined in [1], and in the same time the infimum of T and T * with respect to the star partial order on B(H) (see [11]).…”

“…There are few results from [5] for DR matrices that can be easily proved for CoR operators in the Hilbert space setting. For example, [5, Theorem 4, Theorem 5] are also true for CoR operators, and [5,Theorem 8] can be extended using [11,Theorem 2.4]. However, we can not have elegant characterizations as the one in [5, Theorem 1], since the CoR class is not defined only by mutual positioning of the ranges of appropriate operators.…”

Perturbation estimation for the parallel sum of Hermitian positive semi-definite matrices

“…Ever since the publication of [2], the parallel sum has been studied in the more general settings of non-square matrices under certain conditions of range inclusions [17], of positive operators A and B on a Hilbert space such that the range of A + B is closed [5] and furthermore, without any assumptions on the range of A + B [14,19]. As the generalizations of the parallel sum, shorted operators and the weakly parallel sum are also studied in [1,6,12,16,18] and [7,13], respectively. For many different equivalent definitions and the properties of the parallel sum, see a recent review paper [9] and the references therein.…”