  (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 $B_j=f_j(B)$.
