In mathematics, there are many kinds of inequalities involving matrices and linear operators on Hilbert spaces.
This article covers some important operator inequalities connected with traces of matrices.
denote the set consisting of positive semi-definite
denote the set of positive definite Hermitian matrices.
For operators on an infinite dimensional Hilbert space we require that they be trace class and self-adjoint, in which case similar definitions apply, but we discuss only matrices, for simplicity.
Furthermore, The original proof of this theorem is due to K. Löwner who gave a necessary and sufficient condition for f to be operator monotone.
[5] An elementary proof of the theorem is discussed in [1] and a more general version of it in.
[6] For all Hermitian n×n matrices A and B and all differentiable convex functions
with derivative f ' , or for all positive-definite Hermitian n×n matrices A and B, and all differentiable convex functions f:(0,∞) →
In either case, if f is strictly convex, equality holds if and only if A = B.
A popular choice in applications is f(t) = t log t, see below.
Define By convexity and monotonicity of trace functions,
, which is, and, in fact, the right hand side is monotone decreasing in
yields, which with rearrangement and substitution is Klein's inequality: Note that if
[9] It proves and generalizes a conjecture of E. P. Wigner, M. M. Yanase, and Freeman Dyson.
[11] Six years later other proofs were given by T. Ando [12] and B. Simon,[3] and several more have been given since then.
The most direct proof is due to H. Epstein;[13] see M.B.
Ruskai papers,[14][15] for a review of this argument.
T. Ando's proof [12] of Lieb's concavity theorem led to the following significant complement to it: For all
define the following map For density matrices
is the Umegaki's quantum relative entropy.
[16] The operator version of Jensen's inequality is due to C.
Let f be a continuous function defined on an interval I and let m and n be natural numbers.
If f is convex, we then have the inequality for all (X1, ... , Xn) self-adjoint m × m matrices with spectra contained in I and all (A1, ... , An) of m × m matrices with Conversely, if the above inequality is satisfied for some n and m, where n > 1, then f is convex.
bounded, self-adjoint operators on an arbitrary Hilbert space
E. H. Lieb and W. E. Thirring proved the following inequality in [19] 1976: For any
, Ebadian et al. later extended the inequality to the case where
[25] Von Neumann's trace inequality, named after its originator John von Neumann, states that for any
[27] A simple corollary to this is the following result:[28] For Hermitian
positive semi-definite complex matrices