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]
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,
A function is said to be operator convex if for all and all with eigenvalues in and , the following holds
A function is operator concave if is operator convex;=, that is, 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 is jointly concave if − is jointly convex, i.e. the inequality above for is reversed.
Given a function the associated trace function on is given by
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.
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,
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]
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.
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 ,
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]
Let be a self-adjoint operator such that is trace class. Then for any with
with equality if and only if
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
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, [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.
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.
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 .
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. [16]
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.
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.
For a continuous function defined on an interval the following conditions are equivalent:
for all bounded, self-adjoint operators on an arbitrary Hilbert space with spectra contained in and all on with
every self-adjoint operator with spectrum in .
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
There are several other inequalities close to the Lieb–Thirring inequality, such as the following: [21] for any and
Additionally, building upon the Lieb-Thirring inequality the following inequality was derived: [23] For any and all with , it holds that
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]
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]
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),
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]
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,
A function is said to be operator convex if for all and all with eigenvalues in and , the following holds
A function is operator concave if is operator convex;=, that is, 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 is jointly concave if − is jointly convex, i.e. the inequality above for is reversed.
Given a function the associated trace function on is given by
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.
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,
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]
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.
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 ,
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]
Let be a self-adjoint operator such that is trace class. Then for any with
with equality if and only if
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
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, [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.
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.
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 .
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. [16]
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.
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.
For a continuous function defined on an interval the following conditions are equivalent:
for all bounded, self-adjoint operators on an arbitrary Hilbert space with spectra contained in and all on with
every self-adjoint operator with spectrum in .
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
There are several other inequalities close to the Lieb–Thirring inequality, such as the following: [21] for any and
Additionally, building upon the Lieb-Thirring inequality the following inequality was derived: [23] For any and all with , it holds that
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]
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]
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),