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.
Basic definitions.
Let H"n" denote the space of Hermitian × matrices, H"n"+ denote the set consisting of positive semi-definite × 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 on an interval ⊂ ℝ, one may define a matrix function for any operator with eigenvalues in by defining it on the eigenvalues and corresponding projectors as 
Operator monotone.
A function defined on an interval ⊂ ℝ is said to be operator monotone if ∀, and all with eigenvalues in , the following holds,
where the inequality means that the operator is positive semi-definite. One may check that is, in fact, "not" operator monotone!
Operator convex.
A function formula_4 is said to be operator convex if for all formula_5 and all with eigenvalues in , and formula_6, the following holds
Note that the operator formula_8 has eigenvalues in formula_9, since formula_10 and formula_11 have eigenvalues in .
A function formula_12 is operator concave if formula_13 is operator convex, i.e. the inequality above for formula_12 is reversed.
Joint convexity.
A function formula_15, defined on intervals formula_16 is said to be jointly convex if for all formula_5 and all
formula_18 with eigenvalues in formula_9 and all formula_20 with eigenvalues in formula_21, and any formula_22 the following holds
A function is jointly concave if − is jointly convex, i.e. the inequality above for is reversed.
Trace function.
Given a function : ℝ → ℝ, the associated trace function on H"n" is given by
where has eigenvalues and Tr stands for a trace of the operator.
Convexity and monotonicity of the trace function.
Let : ℝ → ℝ be continuous, and let be any integer. Then, if formula_25 is monotone increasing, so 
is formula_26 on H"n".
Likewise, if formula_27 is convex, so is formula_26 on H"n", and
it is strictly convex if is strictly convex.
See proof and discussion in, for example.
Löwner–Heinz theorem.
For formula_29, the function formula_30 is operator monotone and operator concave.
For formula_31, the function formula_32 is operator monotone and operator concave.
For formula_33, the function formula_32 is operator convex. Furthermore, 
The original proof of this theorem is due to K. Löwner who gave a necessary and sufficient condition for 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 × matrices and and all differentiable convex functions
convex functions :(0,∞) → ℝ, the following inequality holds,
In either case, if is strictly convex, equality holds if and only if = .
A popular choice in applications is , see below.
Proof.
Let formula_37 so that, for formula_38, 
varies from formula_40 to formula_41.
Define
By convexity and monotonicity of trace functions, formula_43 is convex, and so for all formula_38,
which is,
and, in fact, the right hand side is monotone decreasing in formula_47.
Taking the limit formula_48 yields,
which with rearrangement and substitution is Klein's inequality:
Note that if formula_51 is strictly convex and formula_52, then formula_53 is strictly convex. The final assertion follows from this and the fact that formula_54 is monotone decreasing in formula_47.
Golden–Thompson inequality.
In 1965, S. Golden and C.J. Thompson independently discovered that
For any matrices formula_56,
This inequality can be generalized for three operators: for non-negative operators formula_58,
Peierls–Bogoliubov inequality.
Let formula_60 be such that Tr e"R" = 1.
Defining , we have 
The proof of this inequality follows from the above combined with Klein's inequality. Take .
Gibbs variational principle.
Let formula_62 be a self-adjoint operator such that formula_63 is trace class. Then for any formula_64 with formula_65
with equality if and only if formula_67
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 formula_68 matrices formula_69, and all formula_70 and formula_71 such that formula_72 and formula_73, with formula_74 the real valued map on formula_75 given by
Here formula_79 stands for the adjoint operator of formula_80
Lieb's theorem.
For a fixed Hermitian matrix formula_81, the function
is concave on formula_83.
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 formula_84 matrices formula_69, and all formula_86 and formula_87 with formula_88, the real valued map on formula_89 given by
is convex.
Joint convexity of relative entropy.
For two operators formula_91 define the following map
For density matrices formula_93 and formula_94, the map formula_95 is the Umegaki's quantum relative entropy.
Note that the non-negativity of formula_96 follows from Klein's inequality with formula_97.
Statement.
The map formula_98 is jointly convex.
Proof.
For all formula_99, formula_100 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.
Jensen's operator and trace inequalities.
The operator version of Jensen's inequality is due to C. Davis.
A continuous, real function formula_12 on an interval formula_9 satisfies Jensen's Operator Inequality if the following holds
for operators formula_106 with formula_107 and for self-adjoint operators formula_108 with spectrum on formula_9.
See, for the proof of the following two theorems.
Jensen's trace inequality.
Let be a continuous function defined on an interval and let and be natural numbers. If is convex, we then have the inequality
for all (1, ... , "n") self-adjoint × matrices with spectra contained in and
all (1, ... , "n") of × matrices with 
Conversely, if the above inequality is satisfied for some and , where > 1, then is convex.
Jensen's operator inequality.
For a continuous function formula_12 defined on an interval formula_9 the following conditions are equivalent:
for all formula_117 bounded, self-adjoint operators on an arbitrary Hilbert space formula_118 with
spectra contained in formula_9 and all formula_120 on formula_118 with formula_122
every self-adjoint operator formula_126 with spectrum in formula_9.
Araki–Lieb–Thirring inequality.
E. H. Lieb and W. E. Thirring proved the following inequality in in 1976: For any formula_135, formula_136 and formula_137
In 1990 H. Araki generalized the above inequality to the following one: For any formula_139, formula_136 and formula_141
and
The Lieb–Thirring inequality also enjoys the following generalization: for any formula_139, formula_136 and formula_148
Effros's theorem and its extension.
E. Effros in proved the following theorem.
If formula_150 is an operator convex function, and formula_151 and formula_152 are commuting bounded linear operators, i.e. the commutator formula_153, the "perspective"
is jointly convex, i.e. if formula_155 and formula_156 with formula_157 (i=1,2), formula_158,
Ebadian et al. later extended the inequality to the case where formula_151 and formula_152 do not commute .
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 formula_162 and formula_163 respectively,
A simple corollary to this is the following result: For hermitian "n" × "n" positive semidefinite complex matrices "A", "B" where now the eigenvalues are sorted decreasingly (formula_165 and formula_166, respectively),