论文标题
标量和块矩阵的标准最佳因素化
Norm optimal factorizations of scalar and block matrices
论文作者
论文摘要
对于$ m \ times n $复杂矩阵$ x等级$ r $，带有schur乘数$ s_x $，我们表明存在$ r \ times m $ $复杂矩阵$ l $和a $ r \ times n $复杂矩阵$ r $ \ |^{\ frac {1} {2}}} \ | \ mathrm {diag}（r^*r）\ | ^{\ frac {1} {2}}，$，标准条件是最佳的。令$ x $在$（\ mathbb {c}^m，\ |。\ | _ \ infty）\ times（\ mathbb {c}^n，\ |。 $ x =δ（η）^*cδ（ξ）$，$ \ mathbb {c}^m，$ \ $ $ $ $ $ $ $ $ $ $ \ $ \ mathbb {c}^n $，以便外部因素是对角操作员，具有$ \ | em |ξ\ | _2 = \ | _2 = \ |最| $ \ | b_x \ | _ {cb}，$，标准条件是最佳的。也提出了对运营商有价值的schur块乘数的概括。
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.
