Von Neumann's trace inequality

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.

Let H_n denote the space of Hermitian n×n matrices, H_n⁺ denote the set consisting of positive semi-definite n×n Hermitian matrices and H_n⁺⁺ 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.

For any real-valued function f on an interval I ⊂ ℝ, one may define a matrix function f(A) for any operator A ∈ H_n with eigenvalues λ in I by defining it on the eigenvalues and corresponding projectors P as

A function f: I → ℝ defined on an interval I ⊂ ℝ is said to be operator monotone if ∀n, and all A,B ∈ H_n with eigenvalues in I, the following holds,

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