Perron–Frobenius theory of seminorms: a topological approach

Abstract

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 : C n → 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.