Perron–Frobenius theory of seminorms: a topological approach

AbstractFor nonnegative matrices A, the well known Perron–Frobenius theory studies the spectral radius ρ(A). Rump has offered a way to generalize the theory to arbitrary complex matrices. He replaced the usual eigenvalue problem with the equation ∣Ax∣=λ∣x∣ and he replaced ρ(A) by the signed spectral radius, which is the maximum λ that admits a nontrivial solution to that equation. We generalize this notion by replacing the linear transformation A by a map f:Cn→R whose coordinates are seminorms, and we use the same definition of Rump for the signed spectral radius. Many of the features of the Perron–Frobenius theory remain true in this setting. At the center of our discussion there is an alternative theorem relating the inequalities f(x)⩾λ∣x∣ and f(x)