On the numerical radius of operators in Lebesgue spaces

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 $$\sup \{\int |x{|}^{p-1}|Tx|d\mu :x\in L_p(\mu ),\Vert x{\Vert }_p=1\}\ge p^{-\frac{1}{p}}{q}^{-\frac{1}{q}}\Vert T\Vert$$ 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.

14 pages