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.E. Carlen, Trace Inequalities and Quantum Entropy: An Introductory Course, Contemp. Math. 529 (2010) 73–140 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).
Basic definitions Let denote the space of Hermitian matrices, denote the set consisting of positive semi-definite Hermitian matrices and 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 one may define a matrix function for any operator with eigenvalues in by defining it on the eigenvalues and corresponding projectors as
given 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. One may check that is, in fact, not operator monotone!
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 if is operator convex; that is, the inequality above for is reversed.
Joint convexity
A function defined on intervals is said to be if for all and all with eigenvalues in and all with eigenvalues in and any the following holds
A function is 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 Hn.
Likewise, if is convex, so is on Hn, 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 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 to be operator monotone. An elementary proof of the theorem is discussed in and a more general version of it in.W.F. Donoghue, Jr., Monotone Matrix Functions and Analytic Continuation, Springer, (1974).
Klein's inequality For all Hermitian × matrices and and all differentiable convex functions
with derivative , or for all positive-definite Hermitian × matrices and , and all differentiable convex functions :(0,∞) → , 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 so that, for , , varies from to .
Define . By convexity and monotonicity of trace functions, is convex, and so for all , , which is, , and, in fact, the right hand side is monotone decreasing in .
Taking the limit yields, , which with rearrangement and substitution is 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
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 be such that Tr eR = 1. Defining , we have
The proof of this inequality follows from the above combined with Klein's inequality. Take .D. Ruelle, Statistical Mechanics: Rigorous Results, World Scient. (1969).
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 Freeman 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 T. Ando's proof 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.
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.
Jensen's operator and trace inequalities The operator version of Jensen's inequality is due to C. Davis.C. Davis, A Schwarz inequality for convex operator functions, Proc. Amer. Math. Soc. 8, 42–44, (1957).
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, 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 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 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). 1976: For any and
In 1990 H. Araki generalized the above inequality to the following one: For any and
for and
for
There are several other inequalities close to the Lieb–Thirring inequality, such as the following: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). for any and
and even more generally: for any and
The above inequality generalizes the previous one, as can be seen by exchanging by and by with and using the cyclicity of the trace, leading to
Additionally, building upon the Lieb–Thirring inequality the following inequality was derived: V. Bosboom, M. Schlottbom, F. L. Schwenninger, On the unique solvability of radiative transfer equations with polarization, in Journal of Differential Equations, (2024). For any and all with , it holds that
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 and related results
, named after its originator John von Neumann, states that for any complex matrices and with singular values and respectively,
with equality if and only if and share singular vectors.
A simple corollary to this is the following result: For Hermitian positive semi-definite complex matrices and where now the eigenvalues are sorted decreasingly ( and respectively),
See also
References
Scholarpedia primary source.