On the numerical radius of operators in Lebesgue spaces

2010/11/22 by Miguel Martin, Martin, Miguel, Javier Meri +3
Mathematics · #46B04 #46B20 #47A12 #FOS: Mathematics #Functional Analysis (math.FA) #math.FA #msc:46B04 #msc:46B20 #msc:47A12

paper · pdf · doi:10.48550/arxiv.1011.4785

14 pages

arxiv created 2010/11/22 · arxiv updated 2010/11/23

Abstract

We show that the absolute numerical index of the space L_p(μ) is p^-1/p q^-1/q (where 1/p+1/q=1). In other words, we prove that sup\∫ |x|^p-1|Tx| dμ : x∈ L_p(μ), ‖x‖_p=1\ ≥ p^-(1)/(p) q^-(1)/(q) ‖T‖ for every T∈ L(L_p(μ)) and that this inequality is the best possible when the dimension of L_p(μ) 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(μ) for atomless μ when restricting to rank-one operators or narrow operators.

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