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 Basic definitions
- 2 Convexity and monotonicity of the trace function
- 3 Löwner–Heinz theorem
- 4 Klein's inequality
- 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 Joint convexity of relative entropy
- 12 Jensen's operator and trace inequalities
- 13 Araki-Lieb-Thirring inequality
- 14 Effros's theorem and its extension
- 15 Von Neumann's trace inequality
- 16 See also
- 17 References
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
- given the spectral decomposition
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!
A function is said to be operator convex if for all and all A,B ∈ Hn with eigenvalues in I, and , the following holds
Note that the operator has eigenvalues in , since and have eigenvalues in I.
A function is operator concave if is operator convex, i.e. the inequality above for is reversed.
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 g is jointly concave if −g is jointly convex, i.e. the inequality above for g is reversed.
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 is monotone increasing, so is on Hn.
Likewise, if is convex, so is on Hn, and it is strictly convex if f is strictly convex.
See proof and discussion in, for example.
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 f to be operator monotone. An elementary proof of the theorem is discussed in  and a more general version of it in.
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.
Let C = A − B so that, for 0 < t < 1,
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 is monotone decreasing in t.
For any matrices ,
This inequality can be generalized for three operators: for non-negative operators ,
Let be such that Tr eR = 1. Defining g = Tr FeR, we have
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
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 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 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 .
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
Lieb-Thirring inequality also enjoys the following generalization: for any , and
Effros's theorem and its extension
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), ,
Ebadian et. al. later extended the inequality to the case where and do not commute . 
Von Neumann's trace inequality
The equality is achieved when and are simultaneously unitarily diagonalizable (see trace).
- E. Carlen, Trace Inequalities and Quantum Entropy: An Introductory Course, Contemp. Math. 529 (2010) 73--140 doi:10.1090/conm/529/10428
- 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 Entropy 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).
- Z. Allen-Zhu, Y. Lee, L. Orecchia, Using Optimization to Obtain a Width-Independent, Parallel, Simpler, and Faster Positive SDP Solver, in ACM-SIAM Symposium on Discrete Algorithms, 1824-1831 (2016).
- E. Effros, A Matrix Convexity Approach to Some Celebrated Quantum Inequalities, Proc. Natl. Acad. Sci. USA, 106, n.4, 1006–1008 (2009).
- A. Ebadian, I. Nikoufar, and M. Gordjic, "Perspectives of matrix convex functions," Proc. Natl Acad. Sci. USA, 108(18), 7313--7314 (2011)
- Mirsky, L. (December 1975). "A trace inequality of John von Neumann". Monatshefte für Mathematik. 79 (4): 303–306. doi:10.1007/BF01647331.
- Scholarpedia primary source.