Total Dilations

Abstract

(1) Let $A$ be an operator on a space $\mathcal{H}$ of even finite dimension. Then for some decomposition $\mathcal{H}=\mathcal{F}\oplus {\mathcal{F}}^{\perp }$, the compressions of $A$ onto $\mathcal{F}$ and ${\mathcal{F}}^{\perp }$ are unitarily equivalent. (2) Let $\{A_j{\}}_{j=0}^n$ be a family of strictly positive operators on a space $\mathcal{H}$. Then, for some integer $k$, we can dilate each $A_j$ into a positive operator $B_j$ on ${\oplus }^k\mathcal{H}$ 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 ${\oplus }^k\mathcal{H}$ and an increasing function $f_j:(0,\infty )⟶(0,\infty )$ such that $B_j=f_j(B)$.

12 pages