On the numerical radius of operators in Lebesgue spaces

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

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