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
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 operator concave if is operator convex;=, that is, 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 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.
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.[5] An elementary proof of the theorem is discussed in [1] and a more general version of it in.[6]
Klein's inequality
For all Hermitian n×n matrices A and B and all differentiable
convex functions
with
derivativef ' , or for all positive-definite Hermitian n×n matrices A and B, and all differentiable
convex functions f:(0,∞) → , 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 log t, 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 .
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 eR = 1.
Defining g = Tr FeR, we have
The proof of this inequality follows from the above combined with
Klein's inequality. Take 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
Freeman 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
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.
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 .[25]
Von Neumann's trace inequality and related results
Von Neumann's trace inequality, named after its originator
John von Neumann, states that for any complex matrices and with
singular values and respectively,[26]
with equality if and only if and share singular vectors.[27]
A simple corollary to this is the following result:[28] For
Hermitian positive semi-definite complex matrices and where now the
eigenvalues are sorted decreasingly ( and respectively),
^
abB. 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).
^Löwner, Karl (1934). "Über monotone Matrixfunktionen". Mathematische Zeitschrift (in German). 38 (1). Springer Science and Business Media LLC: 177–216.
doi:
10.1007/bf01170633.
ISSN0025-5874.
S2CID121439134.
^D. Ruelle, Statistical Mechanics: Rigorous Results, World Scient. (1969).
^Wigner, Eugene P.; Yanase, Mutsuo M. (1964). "On the Positive Semidefinite Nature of a Certain Matrix Expression". Canadian Journal of Mathematics. 16. Canadian Mathematical Society: 397–406.
doi:
10.4153/cjm-1964-041-x.
ISSN0008-414X.
S2CID124032721.
^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).
^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).
^L. Lafleche, C. Saffirio, Strong Semiclassical Limit from Hartree and
Hartree-Fock to Vlasov-Poisson Equation, arXiv:2003.02926 [math-ph].
^ V. Bosboom, M. Schlottbom, F. L. Schwenninger, On the unique solvability of radiative transfer equations with polarization, in Journal of Differential Equations, (2024).
^Mirsky, L. (December 1975). "A trace inequality of John von Neumann". Monatshefte für Mathematik. 79 (4): 303–306.
doi:
10.1007/BF01647331.
S2CID122252038.
^Carlsson, Marcus (2021). "von Neumann's trace inequality for Hilbert-Schmidt operators". Expositiones Mathematicae. 39 (1): 149–157.
doi:
10.1016/j.exmath.2020.05.001.