Numerical Radii of Disjoint-Support and Finite-Rank Operators on Real L p Spaces

Let X = L p ([0, 1]; R), where 1 < p < ∞ and q = p/(p − 1). We study the numerical radius of operators whose input and output supports are either separated or allowed to overlap. First, we give a direct sharp estimate for operators satisfying T = PBT PA with A ∩ B null: v(T) ≥ κp ∥T∥ , κp = p −1/pq −1/q , and exhibit a rank-one operator attaining equality. Second, we give a detailed finite-rank proof of the lower estimate v(T) ≥ βp ∥T∥ , βp = sup τ>0 κpτ p−1 − τ 1 + τ p , without any disjointness assumption on the representing functionals and range vectors. The proof uses mean-zero signs, the Hobby–Rice theorem, balanced dyadic partitions, and a two-sign cancellation argument. The second estimate is not claimed as a new theorem: it is the finite-rank specialization of the stronger result of Martín, Merí, and Popov, who proved the same lower bound for every narrow operator on an atomless finite measure space. We state this attribution explicitly and distinguish the proved estimates from the unresolved exact-value questions. In particular, the equality between the narrow numerical index and the usual numerical index is not established here.