Inequalities for the Hadamard weighted geometric mean of positive kernel operators on Banach function spaces

Let K-1,..., K-n be positive kernel operators on a Banach function space. We prove that the Hadamard weighted geometric mean of K-1,..., K-n, the operator K, satisfies the following inequalities parallel to K parallel to <= parallel to K-1 parallel to(alpha 1)parallel to K-2 parallel to(alpha 2)...parallel to K-n parallel to(alpha n) and r(K) <= r(K-1) (alpha 1)r(K-2)(alpha 2) ... r(K-n)(alpha n), where parallel to.parallel to and r(.) denote the operator norm and the spectral radius, respectively. In the case of completely atomic measure space we show some additional results. In particular, we prove an infinite-dimensional extension of the known characterization of those functions f : R-+(n) --> R+ satisfying r(f(A(1),..., A(n))) <= f(r(A(1)),..., r(A(n))) for all non-negative matrices A(1),..., A(n) of the same order.