Naomi Steen (Belfast) - "A characterisation of positive local Schur multipliers"

Abstract

Let (X, μ), (Y, ν) be standard σ-finite measure spaces. Identifying L²(X × Y) with C₂(L²(X), L²(Y)) and letting φ ∈ L^∞(X × Y), consider the map S_φ on C₂(L²(X), L²(Y)) given by S_φ(T_k) = T_φk ∈ L²(X × Y). Then φ is called a measurable Schur multiplier if S_φ is bounded in the operator norm. Peller obtained the following characterisation: φ ∈ L^∞(X × Y) is a Schur multiplier if and only if there exist {a_i} ⊆ L^∞(X), {b_i} ⊆ L^∞(Y) and C > 0 such that ∑_i≥1 |a_i(x)|² < C, ∑_i≥1 |b_i(y)|² < C for almost all x ∈ X, y ∈ Y and such that φ (x, y)=∑_i≥1 a_i(x) b_i(y) for almost all (x, y) ∈ X × Y.

Shulman, Todorov and Turowska have more recently introduced and characterised certain unbounded generalisations of these

multipliers, including those termed the local Schur multipliers. We will introduce and characterise the positive local Schur

multipliers, making use of a generalised Stinespring theorem that will also be obtained.