This is an announcement for the paper "On the numerical index of real $L_p(\mu)$-spaces" by Miguel Martin, Javier Meri, and Mikhail Popov.
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 $p\neq 2$. In other words, it is shown that for every bounded linear operator $T$ on the real space $L_p(\mu)$, one has $$ \sup\left{\Bigl|\int |x|^{p-1}\sign(x),T x\ d\mu \Bigr|\ : \ x\in L_p(\mu),,|x|=1\right} \geq \frac{M_p}{8\e}|T| $$ where $\displaystyle M_p=\max_{t\in[0,1]}\frac{|t^{p-1}-t|}{1+t^p}>0$ for every $p\neq 2$. It is also shown that for every bounded linear operator $T$ on the real space $L_p(\mu)$, one has $$ \sup\left{\int |x|^{p-1}|Tx|\ d\mu \ : \ x\in L_p(\mu),,|x|=1\right} \geq \frac{1}{2\e}|T|. $$
Archive classification: math.FA math.OA
Mathematics Subject Classification: 46B04, 46E30, 47A12
The source file(s), MartinMeriPopov.tex: 21471 bytes, is(are) stored in gzipped form as 0903.2704.gz with size 7kb. The corresponding postcript file has gzipped size 74kb.
Submitted from: mmartins@ugr.es
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