Inequalities for the spectral radius of non-negative functions

Let (X, mu) be a sigma-finite measure space and M(X x X)+ a cone of all equivalence classes of (almost everywhere equal) non-negative measurable functions on a product measure space X x X. Using Luxemburg-Gribanov theorem we define a product * on M(X x X)(+). Given a function seminorm h : M(X x X)(+) -> [0, infinity], we introduce the spectral radius r(h)(f) of f is an element of M(X 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.