For the solution of linear systems of equations of the formAu=fby iterations, it is customary to consider a splittingA=M−Nwith the iterative process beingandu0is the initial guess. In this paper we extend the theory of Splittings to Banach spaces and in particular we generalize some results by Schneider, Rose and Szyld on splittings and irreducibility ofM-matrices, and those by Schneider, Neumann and Plemmons on the indices ifAandT. We present a new concept of cone-operator irreducibility which generalizes the usual concept of irreducibility. We also introduce the Concept ofG-compatible spittings forM-operators and show its relation to graph compatible Splittings in the finite dimensional case. Most of our results are graph independent.