Comparison theorems using general cones for norms of iteration matrices

We prove comparison theorems for norms of iteration matrices in splittings of matrices in the setting of proper cones in a finite dimensional real space by considering cone linear absolute norms and cone max norms. Subject to mild additional hypotheses, we show that these comparison theorems can hold only for such norms within the class of cone absolute norms. Finally, in a Banach algebra setting, we prove a comparison theorem for spectral radii without appealing to Perron-Frobenius theory.