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

TL;DR

This paper refines inequalities relating spectral radii of Hadamard geometric means of positive kernel operator sets, extending results to essential spectral radii under certain conditions on the Banach space.

Contribution

It provides new inequalities for spectral and essential spectral radii of Hadamard geometric means of bounded positive kernel operator sets, including cases involving essential spectral radii.

Findings

Refined inequalities for spectral radii of Hadamard geometric means.

Extended inequalities to essential spectral radii under order continuity.

Applicable to Banach spaces with order continuous norms.

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 $L^*$ have order continuous norms.