In mathematics, there are many kinds of inequalities connected with matrices and linear operators on Hilbert spaces. This article reviews some of the most important operator inequalities connected with traces of matrices.
Basic definitions 
Let denote the space of Hermitian matrices and denote the set consisting of positive semi-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 one can define a matrix function for any operator with eigenvalues in by defining it on the eigenvalues and corresponding projectors as with the spectral decomposition
Operator monotone 
A function defined on an interval is said to be operator monotone if for all , and all with eigenvalues in , the following holds:
where the inequality means that the operator is positive semi-definite.
Operator convex 
A function is said to be operator convex if for all and all with eigenvalues in , and , the following holds
Note that the operator has eigenvalues in , since and have eigenvalues in .
A function is operator concave if is operator convex, i.e. the inequality above for is reversed.
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 is jointly concave if is jointly convex, i.e. the inequality above for is reversed.
Trace function 
Given a function , the associated trace function on is given by
where has eigenvalues and stands for a trace of the operator.
Convexity and monotonicity of the trace function 
Let be continuous, and let be any integer.
Then if is monotone increasing, so is on .
Likewise, if is convex, so is on , and
it is strictly convex if is strictly convex.
See proof and discussion in, 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 and operator convex.
Furthermore, is operator concave and operator monotone, while is operator convex.
The original proof of this theorem is due to K. Löwner, where he 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 with derivative , or for all postive-definite Hermitian matrices and , and all differentiable convex functions the following inequality holds
In either case, if is strictly convex, there is equality if and only if .
Let so that for , . Define . By convexity and monotonicity of trace functions, is convex, and so for all ,
and in fact the right hand side is monotone decreasing in . Taking the limit yields Klein's inequality.
Note that if is strictly convex and , then is strictly convex. The final assertion follows from this and the fact that is monotone decreasing in .
Golden–Thompson inequality 
For any matrices ,
This inequality can be generalized for three operators: for non-negative operators ,
Peierls–Bogoliubov inequality 
Let be such that . Define , then
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. 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 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, 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 
For all matrices , and all and with , the real valued map on given by
Joint convexity of relative entropy 
For two operators define the following map
Note that the non-negativity of follows from Klein's inequality with .
The map is jointly convex.
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.
Jensen's operator and trace inequalities 
A continuous, real function on an interval satisfies Jensen's Operator Inequality if the following holds
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 self-adjoin matrices with spectra contained in and all of matrices with .
Conversely, if the above inequality is satisfied for some and , where , then 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  in 1976: For any , and
In 1990  H. Araki generalized the above inequality to the following one: For any , and
Effros's theorem 
E. Effros in  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), ,
See also 
- E. Carlen, Trace Inequalities and Quantum Entropy: An Introductory Course, Contemp. Math. 529 (2009).
- R. Bhatia, Matrix Analysis, Springer, (1997).
- B. Simon, Trace Ideals and their Applications, Cambridge Univ. Press, (1979); Second edition. Amer. Math. Soc., Providence, RI, (2005).
- M. Ohya, D. Petz, Quantum Entropy and Its Use, Springer, (1993).
- K. Löwner, "Uber monotone Matrix funktionen", Math. Z. 38, 177–216, (1934).
- W.F. Donoghue, Jr., Monotone Matrix Functions and Analytic Continuation, Springer, (1974).
- S. Golden, Lower Bounds for Helmholtz Functions, Phys. Rev. 137, B 1127–1128 (1965)
- C.J. Thompson, Inequality with Applications in Statistical Mechanics, J. Math. Phys. 6, 1812–1813, (1965).
- E. H. Lieb, Convex Trace Functions and the Wigner–Yanase–Dyson Conjecture, Advances in Math. 11, 267–288 (1973).
- D. Ruelle, Statistical Mechanics: Rigorous Results, World Scient. (1969).
- E. P. Wigner, M. M. Yanase, On the Positive Semi-Definite Nature of a Certain Matrix Expression, Can. J. Math. 16, 397–406, (1964).
- . Ando, Convexity of Certain Maps on Positive Definite Matrices and Applications to Hadamard Products, Lin. Alg. Appl. 26, 203–241 (1979).
- H. Epstein, Remarks on Two Theorems of E. Lieb, Comm. Math. Phys., 31:317–325, (1973).
- M. B. Ruskai, Inequalities for Quantum Entropy: A Review With Conditions for Equality, J. Math. Phys., 43(9):4358–4375, (2002).
- M. B. Ruskai, Another Short and Elementary Proof of Strong Subadditivity of Quantum Entropy, Reports Math. Phys. 60, 1–12 (2007).
- G. Lindblad, Expectations and Entropyy Inequalities, Commun. Math. Phys. 39, 111–119 (1974).
- C. Davis, A Schwarz inequality for convex operator functions, Proc. Amer. Math. Soc. 8, 42–44, (1957).
- F. Hansen, G. K. Pedersen, Jensen's Operator Inequality, Bull. London Math. Soc. 35 (4): 553–564, (2003).
- E. H. Lieb, W. E. Thirring, Inequalities for the Moments of the Eigenvalues of the Schrödinger Hamiltonian and Their Relation to Sobolev Inequalities, in Studies in Mathematical Physics, edited E. Lieb, B. Simon, and A. Wightman, Princeton University Press, 269-303 (1976).
- H. Araki, On an Inequality of Lieb and Thirring, Lett. Math. Phys. 19, 167-170 (1990).
- E. Effros, A Matrix Convexity Approach to Some Celebrated Quantum Inequalities, Proc. Natl. Acad. Sci. USA, 106, n.4, 1006–1008 (2009).