Norm optimal factorizations of scalar and block matrices

Abstract

For an $m \times n$ complex matrix $X$ of rank $r$ with Schur multiplier $S_X$ we show that there exist an $ r \times m $ complex matrix $L$ and an $ r\times n $ complex matrix $R$ such that $X = L^*R$ and $\|S_X\|\, =\, \|\mathrm{diag} (L^*L) \|^{\frac{1}{2}} \| \mathrm{diag} (R^*R) \| ^{\frac{1}{2}},$ and the norm condition is optimal. Let the completely bounded norm of the bilinear form $B_X$ induced by $X$ on $(\mathbb{C}^m, \|.\|_\infty) \times (\mathbb{C}^n, \|.\|_\infty)$ be denoted $\|B_X\|_{cb},$ then $X$ has a factorization $ X = \Delta(\eta)^* C \Delta(\xi)$ with $\eta $ in $\mathbb{C}^m,$ $\xi$ in $\mathbb{C}^n$ such that the outer factors are diagonal operators with $\|\xi\|_2 = \|\eta\|_2=1 $ and $C$ has operator norm equal to $\|B_X\|_{cb},$ and the norm condition is optimal. A generalization to operator valued Schur block multipliers is presented too.