By Foata D., Han G.-N.

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N. 4 Let A, B be positive semideﬁnite matrices and · be any unitarily invariant norm. Then the following assertions hold. (I) For every nonnegative operator monotone function f (t) on [0, ∞) f (A + B) ≤ f (A) + f (B) . 5) (II) For every nonnegative function g(t) on [0, ∞) with g(0) = 0 and g(∞) = ∞, whose inverse function is operator monotone g(A + B) ≥ g(A) + g(B) . 6) 58 4. Norm Inequalities The main part of the proof is to show successively two lemmas. , the Schatten 2-norm) of a rectangular l × m matrix X = (xij ) : X F ≡ {tr (X ∗ X)}1/2 = {tr (XX ∗ )}1/2 = { |xij |2 }1/2 .

33) for 2 ≤ p ≤ ∞ and 22/p−1 ( A for 1 ≤ p ≤ 2. Proof. When p ≥ 2, f (t) = tp/2 on [0, ∞) is convex and when 1 ≤ p ≤ 2, g(t) = −tp/2 on [0, ∞) is convex. 29) {2−p/2 (s2j + s2n−j+1 )p/2 } ≺w {|αj + iβj |p } {−2−p/2 (s2j + s2n−j+1 )p/2 } ≺w {−|αj + iβj |p } for p ≥ 2, for 1 ≤ p ≤ 2. In particular, we get n n (s2j + s2n−j+1 )p/2 ≤2 p/2 |αj + iβj |p j=1 j=1 n n (s2j + s2n−j+1 )p/2 ≥ 2p/2 j=1 for p ≥ 2, |αj + iβj |p for 1 ≤ p ≤ 2. 35) j=1 Since for ﬁxed nonnegative real numbers a, b, the function t → (at + bt )1/t is decreasing on (0, ∞), spj + spn−j+1 ≤ (s2j + s2n−j+1 )p/2 for p ≥ 2 and this inequality is reversed when 1 ≤ p ≤ 2.

Every matrix T can be written uniquely as T = A + iB with A, B Hermitian: A= T + T∗ , 2 B= T − T∗ . 27) This is the matrix Cartesian decomposition. A and B are called the real and imaginary parts of T. We will study the relations between the eigenvalues of A, B and the singular values of T. 24] on eigenvalues of Hermitian matrices. Note that we always denote the eigenvalues of a Hermitian matrix A in decreasing order: λ1 (A) ≥ · · · ≥ λn (A) and write λ(A) ≡ (λ1 (A), . . , λn (A)). 40 3. 18 (Lidskii) Let G, H ∈ Mn be Hermitian.

### A basis for the right quantum algebra and the “1 = q” principle by Foata D., Han G.-N.

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