We review some convexity inequalities for Hermitian matrices an add one more
to the list.Comment: accepted in LA

Bourin, Jean-christophe

English

arXiv

AbstractWe review some recent convexity results for Hermitian matrices and we add a new one to the list: Let A be semidefinite positive, let Z be expansive, Z∗Z⩾I, and let f:[0,∞)→[0,∞) be a concave function. Then, for all symmetric norms‖f(Z∗AZ)‖⩽‖Z∗f(A)Z‖.This inequality complements a classical trace inequality of Brown–Kosaki

Bourin, Jean-Christophe

Elsevier - Publisher Connector 

A concavity inequality for symmetric norms 

Linear Algebra and its Applications 413 (2006) 212–217
www.elsevier.com/locate/laa
A concavity inequality for symmetric norms
Jean-Christophe Bourin
Dépt. de Mathématiques, Université de Cergy-Pontoise, 2 rue Adolphe Chauvin,
95302 Pontoise, France
Received 8 July 2005; accepted 1 September 2005
Available online 25 October 2005
Submitted by C.-K. Li
Abstract
We review some recent convexity results for Hermitian matrices and we add a new one to
the list: Let A be semidefinite positive, let Z be expansive, Z∗Z  I , and let f : [0,∞) −→
[0,∞) be a concave function. Then, for all symmetric norms
‖f (Z∗AZ)‖  ‖Z∗f (A)Z‖.
This inequality complements a classical trace inequality of Brown–Kosaki.
© 2005 Published by Elsevier Inc.
AMS classification: 47A30; 47A63
Keywords: Hermitian operators; Eigenvalues; Operator inequalities; Jensen’s inequality
Introduction
A good part of matrix analysis consists in establishing results for Hermitian oper-
ators considered as generalized real numbers. In particular several results are matrix
versions of inequalities for convex functions f on the real line, such as
f
(
a + b
2
)
 f (a) + f (b)
2
(1)
for all reals a, b and
E-mail: bourinjc@club-internet.fr
0024-3795/$ - see front matter ( 2005 Published by Elsevier Inc.
doi:10.1016/j.laa.2005.09.007
J.-C. Bourin / Linear Algebra and its Applications 413 (2006) 212–217 213
f (za)  zf (a) (2)
for convex functions f with f (0)  0 and scalars a and z with 0 < z < 1.
In this brief note we first review some recent matrix versions of (1), (2) and next
we give the matrix version of the companion inequality of (2):
f (za)  zf (a) (3)
for concave functions f with f (0)  0 and scalars a and z with 1 < z.
Capital letters A,B, . . . , Z mean n-by-n complex matrices, or operators on a
finite dimensional Hilbert space H; I stands for the identity. When A is positive
semidefinite, respectively positive definite, we write A  0, respectively A > 0.
1. Some known convexity results
The following are well known trace versions of elementary inequalities (1)
and (2).
1.1. von Neumann’s trace inequality: For convex functions f and Hermitians
A, B,
Tr f
(
A + B
2
)
 Tr f (A) + f (B)
2
(4)
equivalently Tr ◦ f is convex on the set of Hermitians.
1.2. Brown–Kosaki’s Trace Inequality [5]: Let f be convex with f (0)  0 and let
A be Hermitian. Then, for all contractions Z,
Tr f (Z∗AZ)  Tr Z∗f (A)Z. (5)
1.3. Hansen–Pedersen’s Trace Inequality [7]: Let f be convex and let {Ai}ni=1 be
Hermitians. Then, for all isometric columns {Zi}ni=1,
Tr f
(∑
i
Z∗i AiZi
)
 Tr
∑
i
Z∗i f (Ai)Zi.
Here isometric column means that
∑
i Z
∗
i Zi = I . Hansen–Pedersen’s result contains
(1) and (2).
When f is convex and monotone, we showed [2] that the above trace inequalities
can be extended to operator inequalities up to a unitary congruence. Equivalently we
have inequalities for eigenvalues. Let us give the precise statements corresponding to
von Neumann and Brown–Kosaki trace inequalities.
1.4. Let A, B be Hermitians and let f be a monotone convex function. Then, there
exists a unitary U such that
f
(
A + B
2
)
 U · f (A) + f (B)
2
· U∗. (6)
1.5. Let A be a Hermitian, let Z be a contraction and let f be a monotone convex
function with f (0)  0. Then, there exists a unitary U such that
f (Z∗AZ)  UZ∗f (A)ZU∗. (7)
214 J.-C. Bourin / Linear Algebra and its Applications 413 (2006) 212–217
Statements 1.4 and 1.5 can break down when the monotony assumption is dropped.
But we recently obtained [4] substitutes involving the mean of two unitary congru-
ences. Let us recall the precise result corresponding to inequalities (1) and (6).
1.6. Let f be a convex function, let A, B be Hermitians and set X = f ({A + B}/2)
and Y = {f (A) + f (B)}/2. Then, there exist unitaries U , V such that
X  UYU
∗ + V YV ∗
2
.
Another substitute of (6) for general convex functionsf would be a positive answer
to the following still open problem [2]: Given Hermitians A, B, can we find unitaries
U , V such that
f
(
A + B
2
)
 Uf (A)U
∗ + Vf (B)V ∗
2
?
We turn to a Brown–Kosaki type inequality involving expansive operators Z, that
is Z∗Z  I . We showed the following trace version of the elementary inequality (3).
1.7. Letf be convex withf (0)  0 and letA  0. Then, for all expansive operators
Z,
Tr f (Z∗AZ)  Tr Z∗f (A)Z. (8)
It is interesting to note [2] that, contrarily to the contractive case (5), the assumption
A  0 cannot be dropped. Also, still contrarily to (5), this result cannot be extended
to eigenvalues inequalities like (7). Nevertheless, we have:
1.8. Let f be nonnegative convex with f (0) = 0, let A  0 and let Z be expansive.
Then, for all symmetric norms
‖f (Z∗AZ)‖  ‖Z∗f (A)Z‖. (9)
Here, by symmetric norm we mean a unitarily invariant one, that is ‖A‖ = ‖UAV ‖
for all operators A and all unitaries U , V .
2. A new concavity result
Of course iff is concave withf (0)  0 then inequality (8) is reversed and provides
an extension of its scalar version (3). Assuming furthermore f nonnegative we tried to
extend it to all symmetric norms but, besides the trace norm, we only got the operator
norm case. Here we may state:
Theorem 2.1. Let f : [0,∞) −→ [0,∞) be a concave function. Let A  0 and let
Z be expansive. Then, for all symmetric norms
‖f (Z∗AZ)‖  ‖Z∗f (A)Z‖.
Proof. It suffices to prove the theorem for the Ky Fan k-norms ‖ · ‖k (cf. [1]). This
shows, since Z is expansive, that we may assume that f (0) = 0. Note that f is
J.-C. Bourin / Linear Algebra and its Applications 413 (2006) 212–217 215
necessarily nondecreasing. Hence, there exists a rank k spectral projection E for
Z∗AZ, corresponding to the k-largest eigenvalues λ1(Z∗AZ), . . . , λk(Z∗AZ) of
Z∗AZ, such that
‖f (Z∗AZ)‖k =
k∑
j=1
λj (Z
∗AZ) = Tr Ef (Z∗AZ)E.
Therefore, using a well known property of Ky Fan norms, it suffices to show that
Tr Ef (Z∗AZ)E  TrEZ∗f (A)ZE.
This is the same as requiring that
Tr EZ∗g(A)ZE  Tr Eg(Z∗AZ)E (10)
for all convex functions g on [0,∞) with g(0) = 0. Any such function can be
approached by a combination of the type
g(t) = λt +
n∑
i=1
αi(t − βi)+ (11)
for a scalar λ and some nonnegative scalars αi and βi . Here (x)+ = max{0, x}. By
using the linearity of the trace it suffices to show that (10) holds for gβ(t) = (t − β)+,
β  0. We claim that there exists a unitary U such that
Z∗gβ(A)Z  Ugβ(Z∗AZ)U∗. (12)
This claim and a basic property of the trace then show that (10) holds for gβ . Indeed,
we then have
TrEZ∗gβ(A)ZE=
k∑
j=1
λj (EZ
∗gβ(A)ZE)

k∑
j=1
λj (Z
∗gβ(A)Z)

k∑
j=1
λj (gβ(Z
∗AZ)) (by (12))
=
k∑
j=1
λj (Egβ(Z
∗AZ)E)
=Tr Egβ(Z∗AZ)E,
where the fourth equality follows from the fact that gβ is nondecreasing and hence E
is also a spectral projection of gβ(Z∗AZ) corresponding to the k largest eigenvalues.
The inequality (12) has been established in [2] in order to prove (8). Let us recall
the proof of (12): We will use the following simple fact. If B is a positive operator with
216 J.-C. Bourin / Linear Algebra and its Applications 413 (2006) 212–217
SpB ⊂ {0} ∪ (x,∞), then we also have SpZ∗BZ ⊂ {0} ∪ (x,∞). Indeed Z∗BZ and
B1/2ZZ∗B1/2 (which is greater than B) have the same spectrum.
Let P be the spectral projection of A corresponding to the eigenvalues strictly
greater than β and let Aβ = AP . Since Z∗AZ − βI  Z∗AβZ − βI and t −→ t+
is nondecreasing, there exists a unitary operator V such that
(Z∗AZ − βI)+  V (Z∗AβZ − βI)+V ∗.
Since Z∗(A − βI)+Z = Z∗(Aβ − βI)+Z we may then assume that A = Aβ . Now,
the above simple fact implies
(Z∗AβZ − βI)+ = Z∗AβZ − βQ,
where Q = suppZ∗AβZ is the support projection of Z∗AβZ. Therefore, using (Aβ −
βI)+ = Aβ − βP , it suffices to show the existence of a unitary operator W such that
Z∗AβZ − βQ  WZ∗(Aβ − βP )ZW ∗ = WZ∗AβZW ∗ − βWZ∗PZW ∗.
But, here we can take W = I . Indeed, we have
suppZ∗PZ = Q (∗) and SpZ∗PZ ⊂ {0} ∪ [1,∞) (∗∗),
where (∗∗) follows from the above simple fact and the identity (∗) from the observation
below with X = P and Y = Aβ .
Observation. If X, Y are two positive operators with suppX = suppY , then for
every operator Z we also have suppZ∗XZ = suppZ∗YZ.
To check this, we establish the corresponding equality for the kernels,
ker Z∗XZ = {h : Zh ∈ ker X1/2} = {h : Zh ∈ ker Y 1/2} = ker Z∗YZ.

In the above proof, the simple idea of approaching convex functions as in (11) was
fruitful. It is also useful to prove (see [2]) the Rotfel’d trace inequality: For concave
functions f with f (0)  0 and A, B  0,
Trf (A + B)  Trf (A) + Trf (B).
If f is convex with f (0)  0 the reverse inequality holds, in particular we have
McCarthy’s inequality
Tr(A + B)p  TrAp + TrBp
for all p > 1.
Remark 2.2. Though scalars inequalities (2) and (3) or their concave anologous hold
for a more general class than convex or concave functions, the corresponding trace
inequalities need the convexity or concavity assumption (cf. [2]). A fortiori, Theorem
2.1 needs the concavity assumption.
Remark 2.3. When f is operator monotone, Theorem 2.1 extends to an operator
inequality which can be rephrased for contractions as follows: For nonnegative oper-
ator monotone functions f on [0,∞), contractions Z and A  0,
J.-C. Bourin / Linear Algebra and its Applications 413 (2006) 212–217 217
Z∗f (A)Z  f (Z∗AZ).
This is the famous Hansen’s inequality [6]. Similarly when f is operator convex,
Hansen–Pedersen’s trace inequality can be extended to an operator inequality [7]
(see also [3]).
Extensions of Theorem 2.1 to infinite dimensional spaces will be considered in a
forthcoming work.
Acknowledgment
The author is grateful to a referee for its valuable comments.
References
[1] R. Bhatia, Matrix Analysis, Springer, Germany, 1996.
[2] J.-C. Bourin, Convexity or concavity inequalities for Hermitian operators, Math. Ineq. Appl. 7 (4)
(2004) 607–620.
[3] J.-C. Bourin, Compressions, Dilations and Matrix Inequalities, RGMIA monograph, Victoria Univer-
sity, Melbourne, 2004, Available from: <http://rgmia.vu.edu.au/monograph>.
[4] J.-C. Bourin, Hermitian operators and convex functions, J. Math. Ineq., in press.
[5] L.G. Brown, H. Kosaki, Jensen’s inequality in semi-finite von Neumann algebras, J. Operator Theory
23 (1990) 3–19.
[6] F. Hansen, An operator inequality, Math. Ann. 246 (1980) 249–250.
[7] F. Hansen, G.K. Pedersen, Jensen’s operator inequality, Bull. London Math. Soc. 35 (2003) 553–564.


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