We deduce the explicit expressions for(P+Q)Dand(PQ)Dof two matricesPandQunder the conditionsP2Q=PQPandQ2P=QPQ. Also, we give the upper bound of|P+QD-PD|2.

In this paper, using some block-operator matrix techniques, we give necessary and sufficient conditions for the reverse order law to hold for {1, 2, 3}-and {1, 2, 4}-inverses of bounded operators on Hilbert spaces. Furthermore, we present some new equivalents of the reverse order law for the Moore-Penrose inverse.

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Abstract:We in this paper de ne the outer-Perron-Frobenius splitting, which is an extension of the pseudoPerron-Frobenius splitting de ned in [A.N. Sushama, K. Premakumari, K.C. Sivakumar, Extensions of Perron-Frobenius splittings and relationships with nonnegative Moore-Penrose inverse, Linear and Multilinear Algebra 63 (2015) 1-11]. We present some criteria for the convergence of the outer-Perron-Frobenius splitting. The ndings of this paper generalize some known results in the literatures.