Norm and Anti-Norm Inequalities for Positive Semi-Definite Matrices

Abstract

Some subadditivity results involving symmetric (unitarily invariant) norms are obtained. For instance, if [Formula: see text] is a polynomial of degree m with non-negative coefficients, then, for all positive operators A, B and all symmetric norms, [Formula: see text] To give parallel superadditivity results, we investigate anti-norms, a class of functionals containing the Schatten q-norms for q ∈ (0, 1] and q < 0. The results are extensions of the Minkowski determinantal inequality. A few estimates for block-matrices are derived. For instance, let f : [0, ∞) → [0, ∞) be concave and p ∈(1, ∞). If fp(t) is superadditive, then [Formula: see text] for all positive m × m matrix A = [aij]. Furthermore, for the normalized trace τ, we consider functions φ(t) and f(t) for which the functional A ↦ φ ◦ τ ◦ f(A) is convex or concave, and obtain a simple analytic criterion.

Unitary orbits of Hermitian operators with convex or concave functions

“…The above proof is much simpler than the original one. For more details and many results on anti-norms and derived anti-norms, often in connection with Theorem 2.1, see [10,11]. Several results in these papers are generalizations of Corollary 2.3.…”

“…There exist also some subadditivity results involving convex functions[10]. For instance, let g(t) = m k=0 a k t k be a polynomial of degree m with all non-negative coefficients.…”

Concavity of certain matrix trace and norm functions

“…We have quite a few examples of symmetric anti-norms on M + n . The following are among important examples [6,7].…”

Concavity of certain matrix trace and norm functions. II

“…for all A, B ∈ M + l , all reals λ ≥ 0 and all unitaries U in M l . This notion is the superadditive version of usual symmetric norms (see [5] for details on anti-norms). The typical example is the Ky Fan k-anti-norm A {k} := k j=1 λ l+1−j (A) for 1 ≤ k ≤ l, the anti-norm version of Ky Fan k-norm A (k) := k j=1 λ j (A), where λ 1 (A) ≥ · · · ≥ λ l (A) are the eigenvalues of A ∈ M + l in decreasing order with multiplicities.…”