A GENERALIZATION OF LEVINGER'S THEOREM TO POSITIVE KERNEL OPERATORS

Abstract

HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button. We prove some inequalities for the spectral radius of positive operators on Banach function spaces. In particular, we prove 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 $[0, \frac{1}{2}]$. We also prove that $\| A + B^* \| \ge 2 \cdot \sqrt{r(A B)}$ for any positive operators $A$ and $B$ on $L^2(X, \mu)$.