Inequalities on the spectral radius, operator norm and numerical radius of Hadamard weighted geometric mean of positive kernel operators

TL;DR

This paper establishes refined inequalities relating the spectral radius, operator norm, and numerical radius for Hadamard geometric means of positive kernel operators, extending previous results in operator theory.

Contribution

The paper generalizes and refines existing inequalities for spectral radius and norms of Hadamard geometric means of positive kernel operators on Banach function spaces.

Findings

Spectral radius of Hadamard geometric mean is bounded by related operator products.

Operator norm inequalities are established for the case of L^2 spaces.

New bounds improve understanding of spectral properties of positive kernel operators.

Abstract

Recently, several authors have proved inequalities on the spectral radius $\rho$, operator norm $\|\cdot\|$ and numerical radius of Hadamard products and ordinary products of non-negative matrices that define operators on sequence spaces, or of Hadamard geometric mean and ordinary products of positive kernel operators on Banach function spaces. In the present article we generalize and refine several of these results. In particular, we show that for a Hadamard geometric mean $A ^{\left( \frac{1}{2} \right)} \circ B ^{\left( \frac{1}{2} \right)}$ of positive kernel operators $A$ and $B$ on a Banach function space $L$, we have $$ \rho \left(A^{(\frac{1}{2})} \circ B^{(\frac{1}{2})} \right) \le \rho \left((AB)^{(\frac{1}{2})} \circ (BA)^{(\frac{1}{2})}\right)^{\frac{1}{2}} \le \rho (AB)^{\frac{1}{2}}.$$ In the special case $L=L^2(X, \mu)$ we also prove that $$\|A^{(\frac{1}{2})} \circ B^{(\frac{1}{2})} \| \le \rho \left ( (A^* B ) ^{(\frac{1}{2})}\circ (B ^* A)^{(\frac{1}{2})}\right)^{\frac{1}{2}} \le \rho (A^* B )^{\frac{1}{2}}.$$