A generalization of Levinger's theorem to positive kernel operators

Roman Drnovšek

Published 2003-04-18Version 1

We prove some inequalities for the spectral radius of positive operators on Banach function spaces. In particular, we show the following extension of Levinger's 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 $\| A + B^* \| \ge 2 \cdot \sqrt{r(A B)}$ for any positive operators $A$ and $B$ on $L^2(X,\mu)$.