Extensions of Three Matrix Inequalities to Semisimple Lie Groups

We give extensions of inequalities of Araki-Lieb-Thirring, Audenaert, and Simon, in the context of semisimple Lie groups.


Introduction
Let Cn×n denote the vector space of n×n complex matrices. A norm ‖| ·‖| on Cn×n is said to be unitarily invariant if ‖| UAV‖| = ‖| A‖| for all unitary U, V ∈ Cn×n. For example, the spectral norm is unitarily invariant. The characterization of unitarily invariant norms in terms of symmetric gauge functions has been given by von Neumann [16] (see [3, p.91
For A ∈ Cn×n, let |A| = (A * A) 1/2 denote the positive semidefinite part of A in the polar decomposition A = U|A|, where U ∈ Cn×n is unitary. Audenaert [2] obtained the following inequality as a generalization of Araki-Lieb-Thirring inequality. If 0 r 1, the inequality is reversed.
The case 0 r 1 in Theorem 1.2 is not stated in [2], but it can be derived by a similar method as for the case r 1. Let |A| ′ = (AA * ) 1/2 = |A * |. So A = |A| ′ V is also a polar decomposition for some unitary V ∈ Cn×n. Since ABA * is Hermitian, (1.2) is equivalent to ‖| |ABA * | ′r ‖| = ‖| |ABA * | r ‖| ‖| |A| r |B| r |A| r ‖| = ‖| |A * | ′r |B| ′r |A * | ′r ‖| . (1. 3) The following result of Simon [11, p.95] is also interesting (see [3, p. Let x = (x 1 , x 2 , . . . , xn) and y = (y 1 , y 2 , . . . , yn) be in R n . Let x ↓ = (x [1] , x [2] , . . . , x [n] ) denote a rearrangement of the components of x such that x [1] x [2] · · · x [n] . We say that x is majorized by y, denoted by We say that x is weakly majorized by y, denoted by x ≺w y, if the equality becomes inequality. An equivalent where conv Sn · x denotes the convex hull of the orbit of x under the action of the symmetric group Sn ( [6,10]). When x and y are nonnegative, we say that x is log majorized by y, denoted by In other words, when x and y are positive, we have x ≺ log y if and only if log x ≺ log y. For X ∈ Cn×n, let s(X) and λ(X) be the vector of singular values of X in decreasing order and the vector of eigenvalues of X whose absolute values are in decreasing order, respectively. If X, Y ∈ Cn×n, then ‖| X‖| ‖| Y‖| for all unitarily invariant norms ‖| · ‖| if and only if s(X) ≺w s(Y), according to Ky Fan's Dominance Theorem (see [3, p.93]). Since both |ABA * | r and |A| r |B| r |A| r are positive semidefinite, Theorem 1.2 amounts to say λ(|ABA * | r ) ≺w λ(|A| r |B| r |A| r ), if r 1. (1.5) We are going to extend Theorem 1.1, Theorem 1.2, and Theorem 1.3 in the context of noncompact connected semisimple Lie groups. We need to be cautious since norm is a concept for vector spaces but not for groups. Indeed, there is another way to express the relationship between the vectors of eigenvalues of |ABA * | r and |A| r |B| r |A| r : If 0 r 1, the above inequalities are reversed.
Proof. It suffices to show that (2) and (3) (2). To establish the converse, we first notice that λ 1 (|ABA * | r ) λ 1 (|A| r |B| r |A| r ). Then we use the kth compound matrix C k (·) [10, p.502-504] argument. Note that if M = UP (where P = (A * A) 1/2 and U unitary) is the polar decomposition of M ∈ Cn×n, so is C k (M) = C k (U)C k (P). Since the kth compound matrix is multiplicative and respects complex conjugate transpose, we have Moreover, for any positive semidefinite Y ∈ Cn×n, the eigenvalues of C k (Y) are the for k = 1, . . . , n. They are equal when k = n because det |ABA * | r = det(|A| r |B| r |A| r ).

Remark 1.5.
It is claimed in [2] that Theorem 1.2 is a generalization of Theorem 1.1, but no proof is given there. We now offer a proof. If A, B ∈ Cn×n are positive semidefinite, so are ABA * = ABA and |A| r |B| r |A| r = A r B r A r . By Theorem 1.4 and Weyl-Horn's theorem [15], there exists a matrix in Cn×n with eigenvalues λ 1 ((ABA) r ) · · · λn((ABA) r ) and singular values λ 1 (A r B r A r ) · · · λn(A r B r A r The group version of Theorem 1.3 takes the following stronger form: for all unitarily invariant norm. Here is a short proof: The equivalent condition (1.6) is the one that suits our extensions. In other words, we will express our group results in terms Kostant's pre-order instead of unitarily invariant norm or weak majorization. Kostant's preorder and log majorization are equivalent when all matrices are nonsingular, as we will see this in Example 2.2.

Main results
Our goal is to establish a generalized form of (1.6) in the context of noncompact connected semisimple Lie groups. To do so, we need to introduce some basic concepts. The reader is referred to [4,8] for the standard notation.
Let G be a noncompact connected semisimple Lie group with Lie algebra g, let Θ: G → G be a nontrivial Lie group isomorphism with Θ 2 being the identity map on G, called a involution, and let K be the fixed point set of Θ, which is an analytic subgroup of G. The differential map dΘ: g → g of Θ has eigenvalues ±1. The eigenspace of dΘ corresponding to 1 is the Lie algebra k of K, and the eigenspace of dΘ corresponding to −1 is an Ad K-invariant subspace p of g complementary to k. Since G is semisimple, the Killing form B on g is nondegenerate. Let Θ be chosen such that B is negative definite on k and positive definite on p. This is equivalent to say that the bilinear form B Θ defined by is an inner product on g. In this case, the decomposition g = k + p is called Cartan decomposition of g, and dΘ is called Cartan involution of g, and Θ is called Cartan involution of G. Then the map p × K → G, (X, k) ↦ → g = e X k, is a diffeomorphism, where e X := exp X is the Lie group exponential map ([4, VI.Theorem 1.1]). Let P := {e X : X ∈ p}. Then every g ∈ G can be uniquely written as g = p(g)k(g) = pk with p = p(g) ∈ P and k = k(g) ∈ K. The decomposition G = PK is called Cartan decomposition of G. Let * : G → G be the diffeomorphism defined by g * = Θ(g −1 ). Then k * = k −1 for k ∈ K and p * = p for p ∈ P. We remark that P is a subset of the fixed point set of *. An element g ∈ G is said to be normal if g and g * commute.
An element X ∈ g is called real semisimple (resp., nilpotent) if ad X is diagonalizable over R (resp., ad X is nilpotent). An element g ∈ G is called hyperbolic (resp., unipotent) if g = exp X for some real semisimple (resp., nilpotent) X ∈ g; in either case X is unique and we write X = log g. An element g ∈ G is called elliptic if Ad g is diagonalizable over C with eigenvalues of modulus 1.
The following important result, due to Kostant [8], is called the complete multiplicative Jordan decomposition, abbreviated as CMJD. Let a be a maximal abelian subspace of p and let A be the analytic subgroup generated by a. We have p = Ad (K)a ([7, p.378]). The Weyl group W of (g, a) acts simply transitively on a, and also on A through the exponential map exp : a → A.
For any real semisimple element X ∈ g, let W(X) denote the set of elements in a that are conjugate to X:

It is known that W(X) is a single
for any irreducible finite dimensional representation π of G, where ρ(π(g)) denotes the spectral radius of the operator π(g).
The following is our first main result, as an extension of Theorem 1.2.
Thus (2.4) is established. The case 0 r 1 is similar.
We remark that similar technique was first used in [13].
The following is an extension of Theorem 1.1 in the context of semisimple Lie groups, i.e., we can obtain the result for GLn(C) by appropriate scaling on the semisimple Lie group SLn(C). Theorem 2.5. Suppose g, h ∈ G and h * = h. If r 1, then for any finite dimensional character χ of G, χ(p r (ghg * )) χ(p r (g * )p r (h)p r (g * )).
If 0 r 1, then the inequality is reversed.
The case 0 r 1 is similar.
Finally we want to extend Theorem 1.3 (in the form of (1.7)) in the context of semisimple Lie groups. The following theorem asserts that a normal element is the "smallest" in its conjugacy class. Theorem 2.6. If g, h ∈ G such that g is normal, then p(g) ≺ p(hgh −1 ). In particular χ(p(g)) χ(p(hgh −1 )) for any finite dimensional character χ of G.
We remark the above relevant results are also true for the Cartan decomposition G = KP with appropriate adjustment.
The following example shows that there are cases in which the pre-order is not necessarily log majorization. Fix a Cartan decomposition g = k + p with that is, the corresponding Cartan involution on g is dΘ(Y) = −Y * for all Y ∈ g and on G is Θ(g) = (g −1 ) * for all g ∈ G. So K = SO(2n), the special orthogonal group. Pick (︃ cosh tn i sinh tn −i sinh tn cosh tn )︃ : t 1 · · · t n−1 |tn| }︁ , and A+ = exp a+, A = exp a, p = Ad (K)a, P = exp p.
Notice that SO 2n (C) ⊂ SL 2n (C) and P is a subset of the set of n × n positive definite matices in SLn(C). Each f ∈ P is K-conjugate to a unique where f 1 · · · f n−1 |fn|. Its eigenvalues are e fi , e −fi , i = 1, . . . , n, which are the singular values of f . We now identify a with R n . With this identification, the Weyl group W acts on a in the following way: (t 1 , . . . , tn) ↦ → (±t σ (1) , . . . , ±t σ(n) ), in which the total number of negative sign is even. By definition, if f , g ∈ P, then f ≺ g means f+ ∈ conv Wg+, where f+, g+ ∈ a+ are described as above. It means that [12] (f 1 , . . . , f n−1 , |fn|) ≺w (g 1 , . . . , g n−1 , |gn|), and in addition if one and only one of fn and gn is negative. This relation has more structure than the majorization that is the pre-order for SLn(C) on the algebra level while the group level description is log majorization.