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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 \(\{T,S\}\)-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.

MSC:

47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
15A09 Theory of matrix inversion and generalized inverses
47A50 Equations and inequalities involving linear operators, with vector unknowns
65F20 Numerical solutions to overdetermined systems, pseudoinverses
65J10 Numerical solutions to equations with linear operators
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