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
-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
-splitting for arbitrary operators on Banach spaces. The index splitting is a partial case of
-splitting. The obtained results extend and generalize various well-known results for square and rectangular complex matrices.