On the numerical radius of operators in Lebesgue spaces

We show that the absolute numerical index of the space Lp(μ)Lp(μ) is p−1pq−1q (where 1p+1q=1). In other words, we prove thatsup{∫|x|p−1|Tx|dμ:x∈Lp(μ),‖x‖p=1}⩾p−1pq−1q‖T‖ for every T∈L(Lp(μ))T∈L(Lp(μ)) and that this inequality is the best possible when the dimension of Lp(μ)Lp(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 Lp(μ)Lp(μ) for atomless μ when restricting to rank-one operators or narrow operators.