On the numerical index of real $L_p(\mu )$-spaces

Abstract

We give a lower bound for the numerical index of the real space $L_p(\mu )$ showing, in particular, that it is non-zero for $p\ne 2$. In other words, it is shown that for every bounded linear operator $T$ on the real space $L_p(\mu )$, one has $$ \sup{\Bigl|\int |x|^{p-1}\sign(x) T x dμ\Bigr| : x\in L_p(μ), \|x\|=1} \geq \frac{M_p}{12\e}\|T\| $$ where $M_p={\max }_{t\in [0,1]}\frac{|t^{p-1}-t|}{1+t^p}>0$ for every $p\ne 2$. It is also shown that for every bounded linear operator $T$ on the real space $L_p(\mu )$, one has $$ \sup{\int |x|^{p-1}|Tx| dμ: x\in L_p(μ), \|x\|=1} \geq \frac{1}{2\e}\|T\|. $$

Revised version, to appear in Israel J. Math