Norm optimal factorizations of scalar and block matrices

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 $X = Δ(η)^* C Δ(ξ)$ with $η$ in $\mathbb{C}^m,$ $ξ$ in $\mathbb{C}^n$ such that the outer factors are diagonal operators with $\|ξ\|_2 = \|η\|_2=1$ 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.

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