By J.-C. Bourin

Best mathematics books

Partial differential equations with Fourier series and BVP

This example-rich reference fosters a gentle transition from ordinary usual differential equations to extra complex options. Asmar's cozy sort and emphasis on purposes make the fabric available even to readers with constrained publicity to issues past calculus. Encourages desktop for illustrating effects and purposes, yet is additionally compatible to be used with no machine entry.

Extra resources for Compressions, Dilations and Matrix Inequalities

Example text

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}.

8 (1957) 42-44. 31 [4] T. Fack, H. Kosaki. Generalized s-numbers of τ -measurable operators, Pacific J. Math. 123, 269-300 (1986) [5] I. C. Gohberg and A. S. , Amer. Math. Soc. trans. (2) 52 (1966) 201-216. [6] F. Hansen, An operator inequality, Math. Ann. 258 (1980) 249-250. [7] F. Hansen and G. K. Pedersen, Jensen’s inequality for operator sand Lowner’s Theorem, Math. Ann. 258 (1982) 229-241. [8] F. Hansen and G. K. Pedersen, Jensen’s operator inequality, Bull. London Math. Soc. 35 (2003) 553-564.

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 .