短运算符和减序
Shorted operators and minus order
摘要：
设$\mathcal{H}$是Hilbert空间，$L(\mathcal{H})$是$\mathcal{H}$上的有界线性算子代数，$W \in L(\mathcal{H})$是正算子.给定$\mathcal{H}$的一个闭子空间$\mathcal{S}$，我们刻画了$W$到$\mathcal{S}$的缩短算子$W_{/ \mathcal{S}}$是某些集合的最大值和下确界，对于负序$\stackrel{-}{\leq}.$。另外，给定$A \in L(\mathcal{H})$，我们研究了以下考虑负序的算子逼近问题：$$ min_{\stackrel{-}{\leq}} \ \{(AX-I)^*W(AX-I) : X \in L(\mathcal{H}), \mbox{ subject to } N(A^*W)\subseteq N(X) \}. $$我们表明，在一定的条件下，短运营商$W_{/R(A)}$（$W$的范围内的$A$）是最小的这个问题，我们的特点的解决方案。
英文摘要：
Let $\mathcal{H}$ be a Hilbert space, $L(\mathcal{H})$ the algebra of bounded linear operators on $\mathcal{H}$ and $W \in L(\mathcal{H})$ a positive operator. Given a closed subspace $\mathcal{S}$ of $\mathcal{H}$, we characterize the shorted operator $W_{/ \mathcal{S}}$ of $W$ to $\mathcal{S}$ as the maximum and as the infimum of certain sets, for the minus order $\stackrel{-}{\leq}.$ Also, given $A \in L(\mathcal{H})$ with closed range, we study the following operator approximation problem considering the minus order: $$ min_{\stackrel{-}{\leq}} \ \{(AX-I)^*W(AX-I) : X \in L(\mathcal{H}), \mbox{ subject to } N(A^*W)\subseteq N(X) \}. $$ We show that, under certain conditions, the shorted operator $W_{/R(A)}$ (of $W$ to the range of $A$) is the minimum of this problem and we characterize the set of solutions.
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