This is an announcement for the paper "On the numerical radius of operators in Lebesgue spaces" by Miguel Martin, Javier Meri and Mikhail Popov.

Abstract: We show that the absolute numerical index of the space $L_p(\mu)$ is $p^{-\frac{1}{p}} q^{-\frac{1}{q}}$ (where $1/p+1/q=1$). In other words, we prove that $$ \sup\left{\int |x|^{p-1}|Tx|, d\mu , : \ x\in L_p(\mu),,|x|_p=1\right} ,\geq ,p^{-\frac{1}{p}} q^{-\frac{1}{q}},|T| $$ 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.

Archive classification: math.FA

Mathematics Subject Classification: 46B04, 46B20, 47A12

Remarks: 14 pages

Submitted from: mmartins@ugr.es

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1011.4785

or

http://arXiv.org/abs/1011.4785