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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 {\it Berman, A.} and {\it Neumann, M.} [SIAM J. Appl. Math. 31, 307-312 (1976; Zbl 0352.65017)] and {\it Berman, A.} and {\it 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 {\it Wei, Y.} [Appl. Math. Comput. 95, 115-124 (1998; Zbl 0942.15003)] introducing the $\{T,S\}$-splitting for arbitrary operators on Banach spaces. The index splitting is a partial case of $\{T,S\}$-splitting. The obtained results extend and generalize various well-known results for square and rectangular complex matrices.

47A05General theory of linear operators
15A09Matrix inversion, generalized inverses
47A50Equations and inequalities involving linear operators, with vector unknowns
65F20Overdetermined systems, pseudoinverses (numerical linear algebra)
65J10Equations with linear operators (numerical methods)