A GENERALIZATION OF LEVINGERS THEOREM TO POSITIVE KERNEL OPERATORS

We prove some inequalities for the spectral radius of positive operators on Banach function spaces. In particular, we prove the following extension of Levingerʹs theorem. Let K be a positive compact kernel operator on L^2(X,(mu)) with the spectral radius r(K). Then the function (phi) defined by (phi)(t) = r(t K + (1-t) K^*) is non-decreasing on [0,1/2]. We also prove that ||A + B^*|| =>2 . sqrt(r(A B)) for any positive operators A and B on L^2(X, (mu)).