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

By J.-C. Bourin

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Johnson, Topics in matrix analysis, Cambrige University Press, 1991. A. Mirman, Numerical range and norm of a linear operator, Trudy Seminara po Funkcional’ nomu Analyzu 10 (1968) 51-55. [8] Y. Nakamura, Numerical range and norm, Math. Japonica 27 (1982) 149-150. 58 Chapter 4 Inequalities for some commuting pairs of positive operators Introduction For all positive operator A and normal operator Z on a separable Hilbert space, the interpolation inequality As ZAt ∞ ≤ ZAs+t ∞, s, t > 0 holds (we denote by · ∞ the usual operator norm).

2] C. Davis, A Shwarz inequality for convex operator functions, Proc. Amer. Math. Soc. 8 (1957) 42-44. [3] F. Hansen, An operator inequality, Math. Ann. 246 (1980) 249-259. [4] F. Hansen and G. K. Pedersen, Jensen’s inequality for operator sand Lowner’s Theorem, Math. Ann. 258 (1982) 229-241. [5] F. Hansen and G. K. Pedersen, Jensen’s operator inequality, Bull. 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.

Let {Aj }nj=0 be hermitian operators on H with A0 ∞ ≤ 1. Then we can totally dilate them into a monotone family of hermitian operators {Bj }nj=0 on ⊕k H, k = 2n , in such a way that B0 ∞ ≤ 1. References [1] R. -C. Bourin, Singular values of compressions, restrictions and dilations, Linear Algebra Appl. 360 (2003) 259-272. -C. Bourin, Total dilations, Linear Algebra Appl. 368 (2003) 159-169. -D. -K. Li, Numerical ranges and dilations, Linear Multilinear Algebra 47 (2000) 35-48. A. R. Johnson, Matrix analysis, Cambrige University Press, 1985.