2022/10/31 by Christensen, Erik
#15A23 #15A60 #15A63 #46L07 #47A30 #47L25 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #Quantum Algebra (math.QA)
paper · doi:10.48550/arxiv.2211.00591
For an m × n complex matrix X of rank r with Schur multiplier SX we show that there exist an r × m complex matrix L and an r× n complex matrix R such that X = L^*R and ‖SX‖ = ‖diag (L^*L) ‖(1)/(2) ‖ diag (R^*R) ‖ (1)/(2), and the norm condition is optimal. Let the completely bounded norm of the bilinear form BX induced by X on (ℂm, ‖.‖_∞) × (ℂn, ‖.‖_∞) be denoted ‖BX‖cb, then X has a factorization X = Δ(η)^* C Δ(ξ) with η in ℂm, ξ in ℂn such that the outer factors are diagonal operators with ‖ξ‖2 = ‖η‖2=1 and C has operator norm equal to ‖BX‖cb, and the norm condition is optimal. A generalization to operator valued Schur block multipliers is presented too.