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][2][3][4]

Basic definitions

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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

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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

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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

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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

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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

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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.

See proof and discussion in,[1] for example.

Löwner–Heinz theorem

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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

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For all Hermitian n×n matrices A and B and all differentiable convex functions   with derivative f ' , 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

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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

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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

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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

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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

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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

 
  • is jointly concave in  
  • is convex in  .

Here   stands for the adjoint operator of  

Lieb's theorem

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For a fixed Hermitian matrix  , the function

 

is concave on  .

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.

Ando's convexity theorem

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T. Ando's proof [12] 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

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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

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The map   is jointly convex.

Proof

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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.[16]

Jensen's operator and trace inequalities

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The operator version of Jensen's inequality is due to C. Davis.[17]

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,[17][18] for the proof of the following two theorems.

Jensen's trace inequality

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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

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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

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E. H. Lieb and W. E. Thirring proved the following inequality in [19] 1976: For any     and    

In 1990 [20] 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:[21] for any     and     and even more generally:[22] 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: [23] For any   and all   with  , it holds that  

Effros's theorem and its extension

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E. Effros in [24] 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 .[25]

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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),  

See also

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References

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  1. ^ a b c E. Carlen, Trace Inequalities and Quantum Entropy: An Introductory Course, Contemp. Math. 529 (2010) 73–140 doi:10.1090/conm/529/10428
  2. ^ R. Bhatia, Matrix Analysis, Springer, (1997).
  3. ^ a b B. Simon, Trace Ideals and their Applications, Cambridge Univ. Press, (1979); Second edition. Amer. Math. Soc., Providence, RI, (2005).
  4. ^ M. Ohya, D. Petz, Quantum Entropy and Its Use, Springer, (1993).
  5. ^ Löwner, Karl (1934). "Über monotone Matrixfunktionen". Mathematische Zeitschrift (in German). 38 (1). Springer Science and Business Media LLC: 177–216. doi:10.1007/bf01170633. ISSN 0025-5874. S2CID 121439134.
  6. ^ W.F. Donoghue, Jr., Monotone Matrix Functions and Analytic Continuation, Springer, (1974).
  7. ^ Golden, Sidney (1965-02-22). "Lower Bounds for the Helmholtz Function". Physical Review. 137 (4B). American Physical Society (APS): B1127–B1128. Bibcode:1965PhRv..137.1127G. doi:10.1103/physrev.137.b1127. ISSN 0031-899X.
  8. ^ Thompson, Colin J. (1965). "Inequality with Applications in Statistical Mechanics". Journal of Mathematical Physics. 6 (11). AIP Publishing: 1812–1813. Bibcode:1965JMP.....6.1812T. doi:10.1063/1.1704727. ISSN 0022-2488.
  9. ^ a b c Lieb, Elliott H (1973). "Convex trace functions and the Wigner-Yanase-Dyson conjecture". Advances in Mathematics. 11 (3): 267–288. doi:10.1016/0001-8708(73)90011-x. ISSN 0001-8708.
  10. ^ D. Ruelle, Statistical Mechanics: Rigorous Results, World Scient. (1969).
  11. ^ 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. ISSN 0008-414X. S2CID 124032721.
  12. ^ a b Ando, T. (1979). "Concavity of certain maps on positive definite matrices and applications to Hadamard products". Linear Algebra and Its Applications. 26. Elsevier BV: 203–241. doi:10.1016/0024-3795(79)90179-4. ISSN 0024-3795.
  13. ^ Epstein, H. (1973). "Remarks on two theorems of E. Lieb". Communications in Mathematical Physics. 31 (4). Springer Science and Business Media LLC: 317–325. Bibcode:1973CMaPh..31..317E. doi:10.1007/bf01646492. ISSN 0010-3616. S2CID 120096681.
  14. ^ Ruskai, Mary Beth (2002). "Inequalities for quantum entropy: A review with conditions for equality". Journal of Mathematical Physics. 43 (9). AIP Publishing: 4358–4375. arXiv:quant-ph/0205064. Bibcode:2002JMP....43.4358R. doi:10.1063/1.1497701. ISSN 0022-2488. S2CID 3051292.
  15. ^ Ruskai, Mary Beth (2007). "Another short and elementary proof of strong subadditivity of quantum entropy". Reports on Mathematical Physics. 60 (1). Elsevier BV: 1–12. arXiv:quant-ph/0604206. Bibcode:2007RpMP...60....1R. doi:10.1016/s0034-4877(07)00019-5. ISSN 0034-4877. S2CID 1432137.
  16. ^ Lindblad, Göran (1974). "Expectations and entropy inequalities for finite quantum systems". Communications in Mathematical Physics. 39 (2). Springer Science and Business Media LLC: 111–119. Bibcode:1974CMaPh..39..111L. doi:10.1007/bf01608390. ISSN 0010-3616. S2CID 120760667.
  17. ^ a b C. Davis, A Schwarz inequality for convex operator functions, Proc. Amer. Math. Soc. 8, 42–44, (1957).
  18. ^ Hansen, Frank; Pedersen, Gert K. (2003-06-09). "Jensen's Operator Inequality". Bulletin of the London Mathematical Society. 35 (4): 553–564. arXiv:math/0204049. doi:10.1112/s0024609303002200. ISSN 0024-6093. S2CID 16581168.
  19. ^ 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).
  20. ^ Araki, Huzihiro (1990). "On an inequality of Lieb and Thirring". Letters in Mathematical Physics. 19 (2). Springer Science and Business Media LLC: 167–170. Bibcode:1990LMaPh..19..167A. doi:10.1007/bf01045887. ISSN 0377-9017. S2CID 119649822.
  21. ^ 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).
  22. ^ L. Lafleche, C. Saffirio, Strong Semiclassical Limit from Hartree and Hartree-Fock to Vlasov-Poisson Equation, arXiv:2003.02926 [math-ph].
  23. ^ V. Bosboom, M. Schlottbom, F. L. Schwenninger, On the unique solvability of radiative transfer equations with polarization, in Journal of Differential Equations, (2024).
  24. ^ Effros, E. G. (2009-01-21). "A matrix convexity approach to some celebrated quantum inequalities". Proceedings of the National Academy of Sciences USA. 106 (4). Proceedings of the National Academy of Sciences: 1006–1008. arXiv:0802.1234. Bibcode:2009PNAS..106.1006E. doi:10.1073/pnas.0807965106. ISSN 0027-8424. PMC 2633548. PMID 19164582.
  25. ^ Ebadian, A.; Nikoufar, I.; Eshaghi Gordji, M. (2011-04-18). "Perspectives of matrix convex functions". Proceedings of the National Academy of Sciences. 108 (18). Proceedings of the National Academy of Sciences USA: 7313–7314. Bibcode:2011PNAS..108.7313E. doi:10.1073/pnas.1102518108. ISSN 0027-8424. PMC 3088602.
  26. ^ Mirsky, L. (December 1975). "A trace inequality of John von Neumann". Monatshefte für Mathematik. 79 (4): 303–306. doi:10.1007/BF01647331. S2CID 122252038.
  27. ^ 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.
  28. ^ Marshall, Albert W.; Olkin, Ingram; Arnold, Barry (2011). Inequalities: Theory of Majorization and Its Applications (2nd ed.). New York: Springer. p. 340-341. ISBN 978-0-387-68276-1.