正定矩阵的Hadamard乘的指数II
Index of Hadamard multiplication by positive matrices II
摘要：
给定一个确定的非负矩阵$A \in M_n (C)$，研究了A：$I(A) = \max \{λ\ge 0 : A\circ B \ge λB$对所有$0\le B\}$的最小指标，其中$A\circ B$表示Hadamard积$(A\circ B)_{ij} = A_{ij} B_{ij}$ .对于$M_n(C)$中的任意酉不变范数N，我们考虑A的N-指标：$I(N,A) = \min\{N(A\circ B) : B\ge 0$和$N(B) = 1 \}$ .如果A有非负项，则$I(A) = I(\| \cdot \|_{sp}, A)$当且仅当存在具有非负项的向量u使得$Au = (1, >..., 1)^T$。我们还表明，$I(\| \cdot \|_{2}, A)= I(\| \cdot \|_{sp}, {\bar A}\circ A)^{1/2}$。当A是对角矩阵或秩为1的矩阵时，对任意酉不变范数N，给出了I（N，A）的计算公式。作为应用，我们找到了Hilbert空间上有界可逆自伴算子S的最佳常数M（S），使得对所有的$0 \le T$都有$\|STS + S^{-1} T S^{-1} \| \ge M(S) \|T\| $.
英文摘要：
Given a definite nonnegative matrix $A \in M_n (C)$, we study the minimal index of A: $I(A) = \max \{λ\ge 0 : A\circ B \ge λB$ for all $0\le B\}$, where $A\circ B$ denotes the Hadamard product $(A\circ B)_{ij} = A_{ij} B_{ij}$. For any unitary invariant norm N in $M_n(C)$, we consider the N-index of A: $I(N,A) = \min\{N(A\circ B) : B\ge 0$ and $N(B) = 1 \}$. If A has nonnegative entries, then $I(A) = I(\| \cdot \|_{sp}, A)$ if and only if there exists a vector u with nonnegative entries such that $Au = (1, >..., 1)^T$. We also show that $I(\| \cdot \|_{2}, A)= I(\| \cdot \|_{sp}, {\bar A}\circ A)^{1/2}$. We give formulae for I(N, A), for an arbitrary unitary invariant norm N, when A is a diagonal matrix or a rank 1 matrix. As an application we find, for a bounded invertible selfadjoint operator S on a Hilbert space, the best constant M(S) such that $\|STS + S^{-1} T S^{-1} \| \ge M(S) \|T\| $ for all $0 \le T$.
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