Total dilations. II. (Q1414150)

[Part I appeared ibid. 368, 159--169 (2003; Zbl 1058.47005), see the preceding review.] The paper under review concerns total dilations on a finite-dimensional or a separable infinite-dimensional Hilbert space \({\mathcal H}\). If \(X\) is an operator acting on \(\bigoplus^k{\mathcal H}\), \(X_{\mathcal H}\) denotes the compression of the first summand of \(X\). The author shows that, given operators \(A> 0\) and \(B> 0\) on \({\mathcal H}\), the following are equivalent: (i) \(A\leq B\); (ii) there exists an operator \(Z> 0\) on a space \({\mathcal F}\supset{\mathcal H}\) such that \(A= (Z_{\mathcal H})^{-1}\) and \(B= (Z^{-1})_{\mathcal H}\); (iii) there exists an operator \(Z> 0\) on a space \({\mathcal F}\supset{\mathcal H}\) such that \(A= (Z_{\mathcal H})^2\) and \(B= (Z^2)_{\mathcal H}\).