On the numerical radius of operators in Lebesgue spaces

TL;DR

This paper determines the exact absolute numerical index of Lebesgue spaces $L_p(u)$, establishes optimal inequalities relating the numerical radius and operator norm, and provides bounds for specific classes of operators.

Contribution

It precisely computes the numerical index of $L_p(u)$ spaces and offers new bounds for the numerical radius versus operator norm for certain operators.

Findings

The absolute numerical index of $L_p(u)$ is $p^{-1/p} q^{-1/q}$.

The established inequality is optimal for spaces of dimension greater than one.

Lower bounds are provided for the equivalence constants between numerical radius and operator norm for rank-one and narrow operators.

Abstract

We show that the absolute numerical index of the space $L_p(\mu)$ is $p^{-1/p} q^{-1/q}$ (where $1/p+1/q=1$). In other words, we prove that $$ \sup\{\int |x|^{p-1}|Tx|\, d\mu \, : \ x\in L_p(\mu),\,\|x\|_p=1\} \,\geq \,p^{-\frac{1}{p}} q^{-\frac{1}{q}}\,\|T\| $$ for every $T\in \mathcal{L}(L_p(\mu))$ and that this inequality is the best possible when the dimension of $L_p(\mu)$ is greater than one. We also give lower bounds for the best constant of equivalence between the numerical radius and the operator norm in $L_p(\mu)$ for atomless $\mu$ when restricting to rank-one operators or narrow operators.