On the numerical index of real L _p (µ)-spaces

Abstract

We give a lower bound for the numerical index of the real space L _p (µ) showing, in particular, that it is non-zero for p ≠ 2. In other words, it is shown that for every bounded linear operator T on the real space L _p (µ), one has

$$\sup \left\{ {|\int {|x{|^{p - 1}}{\rm{sign}}(x)Tx d\mu |:x \in {L_p}\left( \mu \right), ||x|| = 1} } \right\} \ge {{{M_p}} \over {12{\rm{e}}}}||T||$$

where ${M_p} = {\max _{t \in \left[ {0,1} \right]}}{{|{t^{p - 1}} - t|} \over {1 + {t^p}}} > 0$ for every p ≠ 2. It is also shown that for every bounded linear operator T on the real space L _p (µ), one has

$$\sup \left\{ {\int {|x{|^{p - 1}}|Tx| d\mu :x \in {L_p}\left( \mu \right), ||x|| = 1} } \right\} \ge {1 \over {2{\rm{e}}}}||T||$$

.