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

New results have appeared. The size and the nature of these results has made it clear, that the revised article has to be divided into 2 separate articles under new titles