# Compressions, Dilations and Matrix Inequalities by J.-C. Bourin

By J.-C. Bourin

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London Math. Soc. 35 (2003) 553-564. A. R. Johnson, Matrix Analysis, Cambridge Univ. Press, Cambridge, 1985. [7] B. E. Pecaric, A matrix version of the Ky fan generalization of the Kantorovich inequality, Linear and Multilinear Algebra 36 (1994) 217-221. [8] W. F. Stinepring, Positive functions on C ∗ -algebras, Proc. Amer. Math. Soc. 6 (1955) 211-216. 43 Chapter 3 Commuting dilations and Total dilations Introduction The letter H denotes a separable Hilbert space. H can be real or complex, finite or infinite dimensional.

6 entails a reverse inequality, first proved by B. E. 8. (Mond-Pecaric) Let Z > 0 with extremal eigenvalues a and b. Then, for every subspace E, (ZE )−1 ≥ 4ab (Z −1 )E . 7. Proof. Let E be the projection onto E. 6, for every r > 0, there exists x > 0 such that (a + b)2 (Z + rI). 4ab Since t −→ −1/t is operator monotone we deduce EZE + xE ⊥ ≤ (EZE + xE ⊥ )−1 ≥ 4ab (Z + rI)−1 (a + b)2 so that 4ab {(Z + rI)−1 }E (a + b)2 and the result follows by letting r −→ 0. 9. All the previous inequalities are sharp.

474]. In terms of compressions, this means (ZE )−1 ≤ (Z −1 )E (4) 37 for every subspace E and every Z > 0. 6 entails a reverse inequality, first proved by B. E. 8. (Mond-Pecaric) Let Z > 0 with extremal eigenvalues a and b. Then, for every subspace E, (ZE )−1 ≥ 4ab (Z −1 )E . 7. Proof. Let E be the projection onto E. 6, for every r > 0, there exists x > 0 such that (a + b)2 (Z + rI). 4ab Since t −→ −1/t is operator monotone we deduce EZE + xE ⊥ ≤ (EZE + xE ⊥ )−1 ≥ 4ab (Z + rI)−1 (a + b)2 so that 4ab {(Z + rI)−1 }E (a + b)2 and the result follows by letting r −→ 0.