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

TL;DR

This paper establishes bounds on the generalized and joint spectral radii of the Hadamard geometric mean of bounded sets of positive kernel operators, extending spectral radius inequalities in Banach function spaces.

Contribution

It provides new inequalities relating the spectral radii of Hadamard geometric means to the spectral radii of the product sets of positive kernel operators.

Findings

Inequalities for spectral radii of Hadamard geometric means

Bounds connecting geometric mean and product spectral radii

Extension of spectral radius inequalities in Banach spaces

Abstract

Let $\Psi _1, \ldots \Psi _m$ be bounded sets of positive kernel operators on a Banach function space $L$. We prove that for the generalized spectral radius $\rho$ and the joint spectral radius $\hat{\rho}$ the inequalities $$\rho \left(\Psi _1 ^{\left( \frac{1}{m} \right)} \circ \cdots \circ \Psi _m ^{\left( \frac{1}{m} \right)} \right) \le \rho (\Psi _1 \Psi _2 \cdots \Psi _m) ^{\frac{1}{m}},$$ $$\hat{\rho} \left(\Psi _1 ^{\left( \frac{1}{m} \right)} \circ \cdots \circ \Psi _m ^{\left( \frac{1}{m} \right)} \right) \le \hat{\rho} (\Psi _1 \Psi _2 \cdots \Psi _m) ^{\frac{1}{m}}$$ hold, where $\Psi _1 ^{\left( \frac{1}{m} \right)} \circ \cdots \circ \Psi _m ^{\left( \frac{1}{m} \right)}$ denotes the Hadamard (Schur) geometric mean of the sets $\Psi _1, \ldots , \Psi _m$.