On the numerical radius of operators in Lebesgue spaces

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 | T x | d μ : 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.