By J.-C. Bourin

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Consequently A and B can not be dilated into commuting Hermitians with norms arbitrarily close to 1. (2) Now, let P , Q be positive, strict contractions. Then, P = (A + I)/2 and Q = (B + I)/2 for some Hermitian, strict contractions A, B. Therefore, dilating P , Q into a commuting pair of positive, strict contractions is a problem equivalent to that of dilating A, B into a commuting pair of Hermitian, strict contractions. By the preceding example, it may be impossible. For an operator A, its numerical range and its numerical angular range are W (A) = { h, Ah | h = 1} and W (A) = { h, Ah | h ∈ H}.

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Proof. A + iB is an operator whose numerical range lies in the region Γ = {z ∈ C : z = x + iy, ε ≤ x ≤ 1, 1 ≤ y} in which ε = ||A−1 ||−1 . Now, fix p > 1 and let Γp = {z ∈ C : z = t + itp , t > 0}. We observe that W (A + iB) is contained in a triangle whose vertices are three points in Γp . Mirman’s theorem entails that A + iB can be dilated into a normal operator N acting on ⊕3 H with Sp(N ) ⊂ Γp . We then deduce that N = Z + iZ p for some strictly positive operator on ⊕3 H. Therefore A = ZH and B = (Z p )H .