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.
Contents
- Basic definitions
- Operator monotone
- Operator convex
- Joint convexity
- Trace function
- Convexity and monotonicity of the trace function
- LwnerHeinz theorem
- Kleins inequality
- Proof
- GoldenThompson inequality
- PeierlsBogoliubov inequality
- Gibbs variational principle
- Liebs concavity theorem
- Liebs theorem
- Andos convexity theorem
- Joint convexity of relative entropy
- Statement
- Jensens operator and trace inequalities
- Jensens trace inequality
- Jensens operator inequality
- Araki Lieb Thirring inequality
- Effross theorem and its extension
- Von Neumanns trace inequality
- References
Basic definitions
Let Hn denote the space of Hermitian n×n matrices, Hn+ denote the set consisting of positive semi-definite n×n Hermitian matrices and Hn++ 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 ∈ Hn with eigenvalues λ in I by defining it on the eigenvalues and corresponding projectors P as
Operator monotone
A function f: I → ℝ defined on an interval I ⊂ ℝ is said to be operator monotone if ∀n, and all A,B ∈ Hn with eigenvalues in I, the following holds,
where the inequality A ≥ B means that the operator A − B ≥ 0 is positive semi-definite. One may check that f(A)=A2 is, in fact, not monotone!
Operator convex
A function
Note that the operator
A function
Joint convexity
A function
A function g is jointly concave if −g is jointly convex, i.e. the inequality above for g is reversed.
Trace function
Given a function f: ℝ → ℝ, the associated trace function on Hn is given by
where A has eigenvalues λ and Tr stands for a trace of the operator.
Convexity and monotonicity of the trace function
Let f: ℝ → ℝ be continuous, and let n be any integer. Then, if
Likewise, if
See proof and discussion in, for example.
Löwner–Heinz theorem
For
For
For
The original proof of this theorem is due to K. Löwner who gave a necessary and sufficient condition for f to be operator monotone. An elementary proof of the theorem is discussed in and a more general version of it in.
Klein's inequality
For all Hermitian n×n matrices A and B and all differentiable convex functions f: ℝ → ℝ with derivative f ' , or for all positive-definite Hermitian n×n matrices A and B, and all differentiable convex functions f:(0,∞) → ℝ, the following inequality holds,
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 logt, see below.
Proof
Let C = A − B so that, for 0 < t < 1,
Define
By convexity and monotonicity of trace functions, φ is convex, and so for all 0 < t < 1,
and, in fact, the right hand side is monotone decreasing in t. Taking the limit t→0 yields Klein's inequality.
Note that if f is strictly convex and C≠ 0, then φ is strictly convex. The final assertion follows from this and the fact that
Golden–Thompson inequality
In 1965, S. Golden and C.J. Thompson independently discovered that
For any matrices
This inequality can be generalized for three operators: for non-negative operators
Peierls–Bogoliubov inequality
Let
The proof of this inequality follows from the above combined with Klein's inequality. Take f(x)= exp(x), A=R+F, and B=R+g I.
Gibbs variational principle
Let
with equality if and only if
Lieb's concavity theorem
The following theorem was proved by E. H. Lieb in. It proves and generalizes a conjecture of E. P. Wigner, M. M. Yanase and F. J. Dyson. Six years later other proofs were given by T. Ando and B. Simon, and several more have been given since then.
For all
Here
Lieb's theorem
For a fixed Hermitian matrix
is concave on
The theorem and proof are due to E. H. Lieb, Thm 6, where he obtains this theorem as a corollary of Lieb's concavity Theorem. The most direct proof is due to H. Epstein; see M.B. Ruskai papers, for a review of this argument.
Ando's convexity theorem
T. Ando's proof of Lieb's concavity theorem led to the following significant complement to it:
For all
is convex.
Joint convexity of relative entropy
For two operators
For density matrices
Note that the non-negativity of
Statement
The map
Proof
For all
is convex. But
and convexity is preserved in the limit.
The proof is due to G. Lindblad.
Jensen's operator and trace inequalities
The operator version of Jensen's inequality is due to C. Davis.
A continuous, real function
for operators
See, for the proof of the following two theorems.
Jensen's trace inequality
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.
Jensen's operator inequality
For a continuous function
for all
every self-adjoint operator
Araki-Lieb-Thirring inequality
E. H. Lieb and W. E. Thirring proved the following inequality in in 1976: For any
In 1990 H. Araki generalized the above inequality to the following one: For any
and
Lieb-Thirring inequality also enjoys the following generalization: for any
Effros's theorem and its extension
E. Effros in proved the following theorem.
If
is jointly convex, i.e. if
Ebadian et. al. later extended the inequality to the case where
Von Neumann's trace inequality
Von Neumann's trace inequality, named after its originator John von Neumann, states that for any n × n complex matrices A, B with singular values
The equality is achieved when