Total dilations

(1) Let A be an operator on a space of even finite dimension. Then for some decomposition , the compressions of A onto and are unitarily equivalent. (2) Let {Aj}j=0n be a family of strictly positive operators on a space . Then, for some integer k, we can dilate each Aj into a positive operator Bj on in such a way that: (i) The operator diagonal of Bj consists of a repetition of Aj. (ii) There exist a positive operator B on and an increasing function fj:(0,∞)→(0,∞) such that Bj=fj(B).