1. Basic definitions
    Operator monotone  Operator convex   {{anchor|Joint_convexity_function_2016_10}}Joint convexity   Trace function 

2. Convexity and monotonicity of the trace function

3. Löwner–Heinz theorem

4. {{anchor|Klein2016_10}}Klein's inequality
    Proof 

5. Golden–Thompson inequality

6. Peierls–Bogoliubov inequality

7. Gibbs variational principle

8. Lieb's concavity theorem

9. Lieb's theorem

10. Ando's convexity theorem

11. {{anchor|Joint_convexity_2016_10}}Joint convexity of relative entropy
     Statement  Proof 

12. Jensen's operator and trace inequalities
     Jensen's trace inequality  Jensen's operator inequality 

13. Araki–Lieb–Thirring inequality

14. Effros's theorem and its extension

15. Von Neumann's trace inequality and related results

16. See also

17. References

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.^[1][2][3][4]

Basic definitions

Let H_n denote the space of Hermitian {{mvar|n}}×{{mvar|n}} matrices, H_n⁺ denote the set consisting of positive semi-definite {{mvar|n}}×{{mvar|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 {{mvar|f}} on an interval {{mvar|I}} ⊂ ℝ, one may define a matrix function {{math|f(A)}} for any operator {{math|A ∈ H_n}} with eigenvalues {{mvar|λ}} in {{mvar|I}} by defining it on the eigenvalues and corresponding projectors {{mvar|P}} as

given the spectral decomposition

Operator monotone

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

where the inequality {{math|A ≥ B}} means that the operator {{math|A − B ≥ 0}} is positive semi-definite. One may check that {{math|f(A){{=}}A²}} is, in fact, not operator monotone!

Operator convex

A function is said to be operator convex if for all and all {{math|A,B ∈ H_n}} with eigenvalues in {{mvar|I}}, and , the following holds

Note that the operator has eigenvalues in , since and have eigenvalues in {{mvar|I}}.

A function is operator concave if is operator convex, i.e. the inequality above for is reversed.

{{anchor|Joint_convexity_function_2016_10}}Joint convexity

A function , defined on intervals is said to be jointly convex if for all and all

with eigenvalues in and all with eigenvalues in , and any the following holds

A function {{mvar|g}} is jointly concave if −{{mvar|g}} is jointly convex, i.e. the inequality above for {{mvar|g}} is reversed.

Trace function

Given a function {{mvar|f}}: ℝ → ℝ, the associated trace function on H_n is given by

where {{mvar|A}} has eigenvalues {{mvar|λ}} and Tr stands for a trace of the operator.

Convexity and monotonicity of the trace function

Let {{mvar|f}}: ℝ → ℝ be continuous, and let {{mvar|n}} be any integer. Then, if is monotone increasing, so

is on H_n.

Likewise, if is convex, so is on H_n, and

it is strictly convex if {{mvar|f}} is strictly convex.

See proof and discussion in,^[1] for example.

Löwner–Heinz theorem

For , the function is operator monotone and operator concave.

For , the function is operator monotone and operator concave.

For , the function is operator convex. Furthermore,

is operator concave and operator monotone, while

is operator convex.

The original proof of this theorem is due to K. Löwner who gave a necessary and sufficient condition for {{mvar|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]

{{anchor|Klein2016_10}}Klein's inequality

For all Hermitian {{mvar|n}}×{{mvar|n}} matrices {{mvar|A}} and {{mvar|B}} and all differentiable convex functions

convex functions {{mvar|f}}:(0,∞) → ℝ, the following inequality holds,

In either case, if {{mvar|f}} is strictly convex, equality holds if and only if {{mvar|A}} = {{mvar|B}}.

A popular choice in applications is {{math|f(t) {{=}} t log t}}, see below.

Proof

Let {{math|C {{=}} A − B}} so that, for 0 < {{mvar|t}} < 1,

Define

By convexity and monotonicity of trace functions, {{mvar|φ}} is convex, and so for all 0 < {{mvar|t}} < 1,

and, in fact, the right hand side is monotone decreasing in {{mvar|t}}. Taking the limit {{mvar|t}}→0 yields Klein's inequality.

Note that if {{mvar|f}} is strictly convex and {{mvar|C}} ≠ 0, then {{mvar|φ}} is strictly convex. The final assertion follows from this and the fact that is monotone decreasing in {{mvar|t}}.

Golden–Thompson inequality

In 1965, S. Golden ^[7] and C.J. Thompson ^[8] independently discovered that

For any matrices ,

This inequality can be generalized for three operators:^[9] for non-negative operators ,

Peierls–Bogoliubov inequality

Let be such that Tr e^R = 1.

Defining {{math|g {{=}} Tr Fe^R}}, we have

The proof of this inequality follows from the above combined with Klein's inequality. Take {{math|f(x) {{=}} exp(x), A{{=}}R + F, and B {{=}} R + gI}}.^[10]

Gibbs variational principle

Let be a self-adjoint operator such that is trace class. Then for any with

with equality if and only if

Lieb's concavity theorem

The following theorem was proved by E. H. Lieb in.^[9] It proves and generalizes a conjecture of E. P. Wigner, M. M. Yanase and F. J. 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.

For all matrices , and all and such that and , with the real valued map on given by

• is jointly concave in
• is convex in .

Here stands for the adjoint operator of

Lieb's theorem

For a fixed Hermitian matrix , the function

is concave on .

The theorem and proof are due to E. H. Lieb,^[9] Thm 6, where he obtains this theorem as a corollary of Lieb's concavity Theorem.

The most direct proof is due to H. Epstein;^[13] see M.B. Ruskai papers,^[14][15] for a review of this argument.

Ando's convexity theorem

T. Ando's proof ^[12] of Lieb's concavity theorem led to the following significant complement to it:

For all matrices , and all and with , the real valued map on given by

is convex.

{{anchor|Joint_convexity_2016_10}}Joint convexity of relative entropy

For two operators define the following map

For density matrices and , the map is the Umegaki's quantum relative entropy.

Note that the non-negativity of follows from Klein's inequality with .

Statement

The map is jointly convex.

Proof

For all , is jointly concave, by Lieb's concavity theorem, and thus

is convex. But

and convexity is preserved in the limit.

The proof is due to G. Lindblad.^[16]

Jensen's operator and trace inequalities

The operator version of Jensen's inequality is due to C. Davis.^[17]

A continuous, real function on an interval satisfies Jensen's Operator Inequality if the following holds

for operators with and for self-adjoint operators with spectrum on .

See,^[17][18] for the proof of the following two theorems.

Jensen's trace inequality

Let {{mvar|f}} be a continuous function defined on an interval {{mvar|I}} and let {{mvar|m}} and {{mvar|n}} be natural numbers. If {{mvar|f}} is convex, we then have the inequality

for all ({{mvar|X}}₁, ... , {{mvar|X}}_n) self-adjoint {{mvar|m}} × {{mvar|m}} matrices with spectra contained in {{mvar|I}} and

all ({{mvar|A}}₁, ... , {{mvar|A}}_n) of {{mvar|m}} × {{mvar|m}} matrices with

Conversely, if the above inequality is satisfied for some {{mvar|n}} and {{mvar|m}}, where {{mvar|n}} > 1, then {{mvar|f}} is convex.

Jensen's operator inequality

For a continuous function defined on an interval the following conditions are equivalent:

• is operator convex.
• For each natural number we have the inequality

for all bounded, self-adjoint operators on an arbitrary Hilbert space with

spectra contained in and all on with

• for each isometry on an infinite-dimensional Hilbert space and

every self-adjoint operator with spectrum in .

• for each projection on an infinite-dimensional Hilbert space , every self-adjoint operator with spectrum in and every in .

Araki–Lieb–Thirring inequality

E. H. Lieb and W. E. Thirring proved the following inequality in ^[19] in 1976: For any , and

In 1990 ^[20] H. Araki generalized the above inequality to the following one: For any , and

for

and

for

The Lieb–Thirring inequality also enjoys the following generalization:^[21] for any , and

Effros's theorem and its extension

E. Effros in ^[22] proved the following theorem.

If is an operator convex function, and and are commuting bounded linear operators, i.e. the commutator , the perspective

is jointly convex, i.e. if and with (i=1,2), ,

Ebadian et al. later extended the inequality to the case where and do not commute . ^[23]

Von Neumann's trace inequality and related results

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 and respectively,^[24]

A simple corollary to this is the following result^[25]: For hermitian n × n complex matrices A, B where now the eigenvalues are sorted decreasingly ( and , respectively),

See also

• von Neumann entropy
• Lieb–Thirring inequality
• Schur–Horn theorem

References

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