On the numerical index of real L_p(μ)-spaces

2009/03/16 by Miguel Martin, Miguel Martı́n, Martin, Miguel +7
Mathematics · #46B04 #46E30 #47A12 #Advanced Banach Space Theory #Advanced Topics in Algebra #Approximation Theory and Sequence Spaces #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #math.FA #math.OA #msc:46B04 #msc:46E30 #msc:47A12

paper · pdf · doi:10.48550/arxiv.0903.2704

Revised version, to appear in Israel J. Math

openalex publication_date 2009/03/16 · arxiv created 2010/01/29 · arxiv updated 2010/02/08 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28

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|∫ |x|^p-1\sign(x) T x dμ| : x∈ L_p(μ), ‖x‖=1 ≥ (M_p)/(12\e)‖T‖ where M_p=max_t∈[0,1]\frac|t^p-1-t|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∫ |x|^p-1|Tx| dμ: x∈ L_p(μ), ‖x‖=1 ≥ (1)/(2\e)‖T‖.

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