Inequalities on the joint and generalized spectral and essential spectral radius of the Hadamard geometric mean of bounded sets of positive kernel operators

Abstract

Let $\Psi$ and $\Sigma$ be bounded sets of positive kernel operators on a Banach function space $L$. We prove several refinements of the known inequalities $$\rho \left({\Psi }^{\left(\frac{1}{2}\right)}\circ {\Sigma }^{\left(\frac{1}{2}\right)}\right)\le \rho (\Psi \Sigma {)}^{\frac{1}{2}}\mathrm{and}\hat{\rho }\left({\Psi }^{\left(\frac{1}{2}\right)}\circ {\Sigma }^{\left(\frac{1}{2}\right)}\right)\le \hat{\rho }(\Psi \Sigma {)}^{\frac{1}{2}}$$ for the generalized spectral radius $\rho$ and the joint spectral radius $\hat{\rho }$, where ${\Psi }^{\left(\frac{1}{2}\right)}\circ {\Sigma }^{\left(\frac{1}{2}\right)}$ denotes the Hadamard (Schur) geometric mean of the sets $\Psi$ and $\Sigma$. Furthermore, we prove that analogous inequalities hold also for the generalized essential spectral radius and the joint essential spectral radius in the case when $L$ and its Banach dual $L^{\ast }$ have order continuous norms.

arXiv admin note: substantial text overlap with arXiv:1612.01765, arXiv:1612.01767