2017/03/23 by Coine, Clément
#FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.1703.08128
Let (Ω1, F1, μ1) and (Ω2, F2, μ2) be two measure spaces and let 1 ≤ p,q ≤ +∞. We give a definition of Schur multipliers on B(Lp(Ω1), Lq(Ω2)) which extends the definition of classical Schur multipliers on B(ℓp,ℓq). Our main result is a characterization of Schur multipliers in the case 1≤ q ≤ p ≤ +∞. When 1 < q ≤ p < +∞, ϕ∈ L∞(Ω1 × Ω2) is a Schur multiplier on B(Lp(Ω1), Lq(Ω2)) if and only if there are a measure space (a probability space when p≠ q) (Ω,μ), a∈ L∞(μ1, Lp(μ)) and b∈ L∞(μ2, Lq'(μ)) such that, for almost every (s,t) ∈ Ω1 × Ω2, ϕ(s,t)=⟨ a(s), b(t) ⟩. Here, L∞(μ1, Lr(μ)) denotes the Bochner space on Ω1 valued in Lr(μ). This result is new, even in the classical case. As a consequence, we give new inclusion relationships between the spaces of Schur multipliers on B(ℓp,ℓq).