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Splittings of operators and generalized inverses. (English) Zbl 0981.47001
Summary: We extend the notion of the proper splitting of rectangular matrices introduced and investigated in Berman, A. and Neumann, M. [SIAM J. Appl. Math. 31, 307-312 (1976; Zbl 0352.65017)] and Berman, A. and Plemmons, R. J. [SIAM J. Numer. Anal. 11, 145-154 (1974; Zbl 0273.65029)] to $g$-invertible operators on Banach spaces. Also, we extend and generalize the notion of the index splitting of square matrices introduced and investigated in Wei, Y. [Appl. Math. Comput. 95, 115-124 (1998; Zbl 0942.15003)] introducing the $\left\{T,S\right\}$-splitting for arbitrary operators on Banach spaces. The index splitting is a partial case of $\left\{T,S\right\}$-splitting. The obtained results extend and generalize various well-known results for square and rectangular complex matrices.

##### MSC:
 47A05 General theory of linear operators 15A09 Matrix inversion, generalized inverses 47A50 Equations and inequalities involving linear operators, with vector unknowns 65F20 Overdetermined systems, pseudoinverses (numerical linear algebra) 65J10 Equations with linear operators (numerical methods)