总扩张
Total Dilations
摘要：
(1)设$A$是偶数有限维空间${\cal H}$上的算子.然后对某些分解${\cal H}={\cal F}\oplus{\cal F}^{\perp}$，$A$到${\cal F}$和${\cal F}^{\perp}$的压缩是酉等价的. (2)设$\{A_j\}_{j=0}^n$是空间${\cal H}$上的严格正算子族.然后，对于某个整数$k$，我们可以将每个$A_j$扩张为$\oplus^k{\cal H}$上的正算子$B_j$，使得：（i）$B_j$的算子对角线由$A_j$的重复组成。(ii)在$\oplus^k{\cal H}$上存在正算子$B$和增函数$f_j : (0,\infty)\longrightarrow(0,\infty)$使得$B_j=f_j(B)$ .
英文摘要：
(1) Let $A$ be an operator on a space ${\cal H}$ of even finite dimension. Then for some decomposition ${\cal H}={\cal F}\oplus{\cal F}^{\perp}$, the compressions of $A$ onto ${\cal F}$ and ${\cal F}^{\perp}$ are unitarily equivalent. (2) Let $\{A_j\}_{j=0}^n$ be a family of strictly positive operators on a space ${\cal H}$. Then, for some integer $k$, we can dilate each $A_j$ into a positive operator $B_j$ on $\oplus^k{\cal H}$ in such a way that: (i) The operator diagonal of $B_j$ consists of a repetition of $A_j$. (ii) There exist a positive operator $B$ on $\oplus^k{\cal H}$ and an increasing function $f_j : (0,\infty)\longrightarrow(0,\infty)$ such that $B_j=f_j(B)$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
