Inequalities for the spectral radius of non-negative functions

Abstract

Let (X, μ) be a σ-finite measure space and M(X × X)+ a cone of all equivalence classes of (almost everywhere equal) non-negative measurable functions on a product measure space X × X. Using Luxemburg-Gribanov theorem we define a product * on M(X × X)+. Given a function seminorm h : M(X × X)+ → [0, ∞], we introduce the spectral radius r h (f) of f ∈ M(X × X)+ with respect to h and *. We give several examples. In particular, r h (f) provides a generalization and unification of the spectral radius and its max version, the maximum cycle geometric mean, of a non-negative matrix.