# On the numerical radius of operators in Lebesgue spaces - Mathematics > Functional Analysis

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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 thedimension of $L p\mu$ is greater than one. We also give lower bounds for thebest constant of equivalence between the numerical radius and the operator normin $L p\mu$ for atomless $\mu$ when restricting to rank-one operators ornarrow operators.

Autor: Miguel Martin Granada, Javier Meri Granada, Mikhail Popov Chernivtsi

Fuente: https://arxiv.org/