Perron-Frobenius theory of seminorms: a topological approach

For 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) < λ∣x∣, which follows from topological principals. This enables us to free the theory from matrix theoretic considerations and discuss it in the generality of seminorms. Some consequences for P-matrices and D-stable matrices are discussed.