Norm optimal factorizations of scalar and block matrices (2211.00591v2)

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.