数理解析研究所講究録
第 1678 巻 2010 年 40-46401

CONCAVE FUNCTIONS OF POSITIVE OPERATORS, SUMS ANDCONGRUENCES
JEAN-CHRISTOPHE BOURIN and EUN-YOUNG LEE
be positive semidefinite matrices of same size and suppose
ABSTRACT. Let , B.
is expansive, i.e., $Z\geq I$ . Two remarkable inequalities are$\Vert f(A+B)\Vert\leq\Vert f(A)+f(B)\Vert$
and $\Vert f(ZAZ)\Vert\leq\Vert Zf(A)Z\Vert$(inand all svmmetric norms
on .
for all non-negative concave functionandresults
surveyseveral relatedparticular for all Schatten p-norms). In this paper wewe show that these inequalities are two aspects of a unique theorem. For the operator
norm, our result also bolds for operators on an infiuiite dimensional Hilbert space.
$A$

$Z$

$Z$

$[0$

$f$

$\Vert\cdot||$$\propto)$

INTRODUCTION
A good part of Matrix Analysis consists in establishing results for Hermitian operators
considered as generalized real numbers or functions. In Section 1 we recall two recent
norm inequalities which are matrix versions of the obvious scalars inequalities
and$f(za)\leq zf(a)$$f(a+b)\leq f(a)+f(b)$. In
and scalars . $b\geq 0$ andon .
for non-negative concave functions
thearenorms
considered
Section 2 we unif.y and generalize these norm inequalities. The
symmetric (or unitarily invariant) norms. Such norms satisfy $\Vert A\Vert=\Vert UAV\Vert$ for all andall unitaries U. . Here and in the sequel capital letters , B. . . . , $Z$ mean n-by-n complex
is positive (semi-definite),matrices, or operators on an n-dimensional Hilbert space. If$Z^{*}Z$dominates the identity , weresp. positive definite. we write $A\geq 0$ , resp. $A>0$ . If
$Z$ is expansive. As a corollary of Section 2 we have:sav that$f$$[0$$a$$\infty)$$\approx\geq 1$

$A$

$A$

$V$

$A$

$I$

be positive and let
Theorem 0.1. Let,andall$p>1$
norms
metric$\{A_{i}\}_{\dot{\iota}=1}^{rn}$$\{Z_{i}\}_{i=1}^{\mathfrak{s}\mathfrak{n}}$be expansive. Then, for all sym-$\Vert\sum z;A_{i}^{p}Z_{i}\Vert\leq\Vert(\sum ZfA_{i}Z_{i})^{p}\Vert$.

The sum is over the first $m$ integers. If $Z_{i}=I$ for all . this is a famous result of Ando andZhan [1] and of Bhatia and Kittaneh [3] in case of integer exponents. The very special
case $Tr(A_{1}^{P}+A_{2}^{p})\leq Tr(A_{1}+A_{2})^{p}$ is Mc-Carthy’s inequalitv [13, p. 20].
$i$

1. Two NORM INEQUALITIESThe cone of positive operators is invariant under congruences $Aarrow S^{*}AS$ and thereare several inequalities involving congruences with a contraction $Z$ and concave functions
AMS subject $C^{t}1assific\cdot ation$ : $15A60,47A30$ .
Key words: Hermitian operators, symmetric norms, operator inequalities.
$1\underline{)}\{K)0$41

JEAN-CHRISTOPHE BOURIN and EUN-YOUNG LEE$f$: [ $0_{1}\infty)arrow[0, \infty)$ . Brown-Kosaki’s trace inequality states(1.1)$Trf(Z^{*}AZ)\geq TrZ^{*}f(A)Z$

Actually a stronger result holds [8] (see also [4]), there exists a unitary $V$ such that(1.2)$f(Z^{*}AZ)\geq VZ^{*}f(A)ZV^{*}$ .

This means that the eigenvalues of $f(Z^{*}AZ)$ dominate those of $Z^{*}f(A)Z$ . Further, if
operator concave (equivalently operator monotone [10]), then Hansen’s operator inequality
holds [9, 10],(1.3)
$f(Z^{*}AZ)\geq Z^{*}f(A)Z^{*}$ .
$f$What happens to (1.1), (1.2), (1.3) when $Z$ is no longer contractive but, in a opposite way,
is expansive ? It is obvious that (1.3) is reversed, meanwhile (1.2) can not be reversed
though a non-trivial proof [4] shows that (1.1) is reversed:(1.4)$Trf(Z^{*}AZ)\leq TrZ^{*}f(A)Z$.

However a quite unexpected phenomena occurs: (1.1) can be extended to all Hermitians
and all concave functions on the real line with $f(O)\geq 0$ , but in (1.4) the assumption$A\geq 0$ can not be dropped.
The good statement for expansive congruences requires. We have [5]:positivity and involves symmetric (or unitarily invariant) norms

$A$

$f$

$\Vert\cdot\Vert$

Theorem 1.1. Let : $[0, \infty)arrow[0, \infty)$ be a concave function. Let $A\geq 0$ and let $Z$ beexpansive. Then, for all symmetric norms,$f$$\Vert f(Z^{*}AZ)\Vert\leq\Vert Z^{*}f(A)Z\Vert$.

Besides these inequalities for congruences, there are nice subadditivity results for concave functions : $[0, \infty)arrow[0, \infty)$ and sums of positive operators. The most elementary
one is a trace inequality companion to (1.1) credited to Rotfel’d [12]: For $A,$ $B\geq 0$ ,$f$

(1.5)

$Trf(A+B)\leq Trf(A)+f(B)$ .

The operator inequality $f(A+B)\leq f(A)+f(B)$ may not hold even if is operator
concave. However, a remarkable subadditivity inequality related to (1.2) is shown in [2]:such that
There exist unitaries $U,$
$f$

$V$

$f(A+B)\leq Uf(A)U^{*}+Vf(B)V^{*}$ .

(1.6)

From this, it follows that the map$xarrow\Vert f(|X|)\Vert$

is subadditive, see [7]. This was first noted by Uchiyama [14]. In case of the trace norm,
this extension of (1.5) is Rotfel’d’s theorem [12]. Of course (1.6) considerably strenghtens(1.5). Another improvement of (1.5), companion to Theorem 0.1, is shown in [7]:Theorem 1.2. Let $A,$ $B\geq 0$ and let
for all symmetric norms,$f$: $[0, \infty)arrow[0, \infty)$ be a concave function. Then,$\Vert f(A+B)\Vert\leq\Vert f(A)+f(B)\Vert$ .

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CONCAVE FUNCTIONS OF POSITIVE OPERATORS, SUMS AND CONGRUENCES

In case of the operator norm, Kosem [11] gave a short proof. The general case is considerably more difficult: When is operator $conc\cdot ave$ , Theorem 1.3 had first $|)een$ proved byAndo and Zhan [1] by using integral representation of operator concave functions and adelicate process. The proof given in [7] is much more elementary.
$f$

2. COMBINED RESULT AND PROOF
We can naturally embodied Theorem 1.1 and Theorem 1.2 (and its version for sums ofseveral operators) in a unique statement:

Theorem 2.1. Let

be positive, letbe expansive and let $f(t)$ be a nonnegative concave function on $[0, \infty)$ . Then, for all symmetrt norms,$\{A_{i}\}_{i=1}^{m}$$\{Z_{i}\}_{i=1}^{m}$$c$$\Vert f(\sum Z_{i}^{*}A_{i}Z_{i})\Vert\leq\Vert\sum Z_{i}^{*}f(A_{i})Z_{i}\Vert$.

Of course the sum is over the first $m$ integers. For $m=1$ we get Theorem 1.1 and if
for all we get Theorem 1.2. There is no obvious way to derive Theorem 2.1
from Theorems 1.1, 1.2. However our proof is adapted from the proof of Theorem 1.1
which is itself partially based on the proof of Theorem 1.2. Therefore our proof is ratherelementary. We use Hansen’s inequality (1.3) for a quite elementary case equivalent tothe fact that $tarrow 1/t$ is operator convex on $(0, \infty)$ . We also use some basic facts about
symmetric norms. In particular we need the following two facts for arbitrary $X,$ $Y\geq 0$ .
1 If
for all symmetric norms, then we also have $\Vert g(X)\Vert\leq\Vert g(Y)\Vert$ for all
symmetric norms and all increasing convex functions $g:[0, \infty)arrow[0, \infty)$ .
2 If
for all Ky Fan k-norms, then we also havefor all symmetric
Inorms (Ky Fan’s principle). Recall that the Ky Fan k-norms are the sum of the largest
singular values.
For this background we refer to any expository text such as [13].
$Z_{i}=I$$i$$)$$\Vert X\Vert\leq\Vert Y\Vert$$)$$\Vert X\Vert_{k}\leq\Vert Y\Vert_{k}$$\Vert X$$\leq\Vert Y\Vert$$k$

Proof of Theorem 2.1. The proof for an arbitrary $m$ is the same than the proof for $m=2$ ,so we consider the case of two expansive operators $X,$
and two positive operators ,$Y$

$B$

$A$

. The proof is divided in four steps.
Step 1. If(1.3):$f$is operator concave, the proof immediately follows from Theorem 1.1 and$\Vert f(X^{*}AX+Y^{*}BY)$ $\Vert\leq\Vert f(X^{*}AX)+f(Y^{*}BY)$$\Vert$$\leq\Vert X^{*}f(A)X+Y^{*}f(B)Y\Vert$ .

(2.1)

Step 2. Now consider a one to one convex function $g:[0, \infty)arrow[0, \infty)$ whose inverse
function is operator concave. Since is onto there exist $A’,$ $B’\geq 0$ such that $A=g(A’)$
and $B=g(B’)$ ; moreover $A’$ and $B’$ can be chosen arbitrarily so that (2.1) can be read as
$f$

$g$

$\Vert f(X^{*}g(A)X+Y^{*}g(B)Y)\Vert\leq\Vert X^{*}AX+Y^{*}BY\Vert$ .

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JEAN-CHRISTOPHE BOURIN and EUN-YOUNG LEE

Since

$g$

is convex increasing we infer(2.2)$\Vert X^{*}g(A)X+Y^{*}g(B)Y\Vert\leq\Vert g(X^{*}AX+Y^{*}BY)\Vert$ .

Step 3. Now we extend (2.2) to the class of all non-negative convex functions onvanishing at . It suffices to consider the Ky Fan k-norms. Suppose thatandboth satisfy (2.2). Using the triangle inequality and the fact that
andare nondecreasing,$[0, \infty)$$0$$\Vert\cdot\Vert_{k}$$g_{2}$$g_{1}$$g_{1}$$g_{2}$$\Vert X^{*}(g_{1}+g_{2})(A)X+Y^{*}(g_{1}+g_{2})(B)Y\Vert_{k}$$\leq\Vert X^{*}g_{1}(A)X+Y^{*}g_{1}(B)Y\Vert_{k}+\Vert X^{*}g_{2}(A)X+Y^{*}g_{2}(B)Y\Vert_{k}$$\leq\Vert g_{1}(X^{*}AX+Y^{*}BY)\Vert_{k}+\Vert g_{2}(X^{*}AX+Y^{*}BY)$$\Vert_{k}$$=\Vert(g_{1}+g_{2})(X^{*}AX+Y^{*}BY)\Vert_{k}$ ,hence the set of functions satisfying to (2.2) is a cone. It is also closed for point-wise
convergence. Since any positive convex function vanishing at can be approached by apositive combination of angle functions at $a>0$ ,$0$$\gamma(t)=\frac{1}{2}\{|t-a|+t-a\}$,it suffices to prove (2.2) for such a . By Step 2 it suffices to approach
whose inverses are operator concave. We take (with $r>0$ )$\gamma$$h_{r}(t)= \frac{1}{2}\{\sqrt{(t-a)^{2}+r}+t-\sqrt{a^{2}+r}\}$$\gamma$by functions,

whose inverse$t- \frac{r/2}{2t+\sqrt{a^{2}+r}-a}+\frac{\sqrt{a^{2}+r}+a}{2}$is operator concave since $1/t$ is operator convex on the positive half-line. Clearly, asconverges uniformly to .

$rarrow 0,$$h_{r}(t)$$\gamma$

Step 4. Proof for any concave function: $[0, \infty)arrow[0, \infty)$ . Again, it suffices
to consider the Ky Fan k-norms. This shows that we may and do assume $f(O)=0$ .
Note thatis necessarily non-decreasing. Hence, there exists a rank
spectral pro$E$
jectionfor $X^{*}AX+Y^{*}BY,$ corresponding to the k-largest eigenvalues $\lambda_{1}(X^{*}AX+$$Y^{*}BY),$$\lambda_{k}(X^{*}AX+Y^{*}BY)$ of $X^{*}AX+Y^{*}BY$ , such that
$f$

$f$

$k$

$\ldots,$$\Vert f(X^{*}AX+Y^{*}BY)$ $\Vert_{k}=\sum_{j=1}^{k}\lambda_{j}(f(X^{*}AX+Y^{*}BY))=$ Tr $Ef(X^{*}AX+Y^{*}BY)E$ .

Therefore, using a well-known property of Ky Fan norms, it suffices to show that

Tr $Ef(X^{*}AX+Y^{*}BY)E\leq$ Tr $E(X^{*}f(A)X+Y^{*}f(B)Y)E$ .
This is the same as requiring that
Tr $E(X^{*}g(A)X+Y^{*}g(B)Y)E\leq$ Tr $Eg(X^{*}AX+Y^{*}BY)E$(2.3)

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CONCAVE FUNCTIONS OF POSITIVE OPERATORS, SUMS AND CONGRUENCES

for all non-positive convex functions on
approached by a combination of the type$g$$[0, \infty)$

with $g(O)=0$ . Any such function can be$g(t)=\lambda t+h(t)$for a scalar $\lambda<0$ and some non-negative convex function
suffices to show that (2.3) holds for $h(t)$ . We have$h$vanishing at$0$. Hence, it

Tr $E(X^{*}h(A)X+Y^{*}h(B)Y)E= \sum_{j=1}^{k}\lambda_{j}(E(X^{*}h(A)X+Y^{*}h(B)Y)E)$$\leq\sum_{j=1}^{k}\lambda_{j}(X^{*}h(A)X+Y^{*}h(B)Y)$$\leq\sum_{j=1}^{k}\lambda_{j}(h(X^{*}AX+Y^{*}BY))$(by Step 3)$= \sum_{j=1}^{k}\lambda_{j}(Eh(X^{*}AX+Y^{*}BY)E)$$=$

Tr $Eh(X^{*}AX+Y^{*}BY)E$

where the second equality follows from the fact that is non-decreasing and hence $E$ is alsoa spectral projection of $h(X^{*}AX+Y^{*}BY)$ corresponding to the largest eigenvalues.
$h$

$k$

$\square$Corollary 2.2. Let $g:[0, \infty)arrow[0, \infty)$ be a convex function with $g(O)=0$ . Let
be positive and let
be expansive. Then, for all symmetric norms,$\{A_{i}\}_{i=1}^{m}$$\{Z_{i}\}_{i=1}^{m}$$\Vert\sum Z_{i}^{*}g(A_{i})Z_{i}\Vert\leq\Vert g(\sum Z_{i}^{*}A_{i}Z_{i})\Vert$.

This corollary is proved in step 3. It can also be derived from Theorem 2.1 by using thefirst fact recalled before the proof. When $Z_{i}=I$ for all this is a remarkable result ofKosem [11]. Note that Theorem 0.1 is a special case of Corollary 2.2. There are several
well-known results involving sums of congruences. But, in contrast with Theorem 2.1 andits corollary, these results deal with strong contractive assumptions. We give an example
generalizing (1.2):$i$Let$\{A_{i}\}_{i=1}^{m}$function onbe positive and$\{Z_{i}\}_{i=1}^{m}$$f(O)\geq 0$ ,$[0, \infty),$such that$\sum Z_{i}^{*}Z_{i}\leq I.$then,If$f( \sum Z_{i}^{*}A_{i}Z_{i})\geq V(\sum Z_{i}^{*}f(A_{i})Z_{i})V^{*}$

for some unitary$V$

is a monotone concave

$f$.

.

Such an inequality is connected to Jensen type inequalities for compressions or positive
unital linear maps, see [4], [2]. We conclude with a discussion of the extension of our resultsfor operators on an infinite dimensional Hilbert space . In the proof of Theorem 2.1, (2.1)
is derived from a version of Hansen’s inequality (1.3) involving congruences with expansive$\mathcal{H}$45

JEAN-CHRISTOPHE BOURIN and EUN-YOUNG LEE

operators $X,$ $Y$ . This Hansen’s inequality remains valid in the infinite dimensional setting
if one requires that $X,$ $Y$ are both expansive and invertible. Consequently we have the.
following result for the usual operator norm$\Vert\cdot\Vert_{\infty}$Theorem 2.1 and its corollaries are still valid forare expansive and invertible.

$\Vert\cdot\Vert_{\infty}$

and opemtors on$\mathcal{H}$when $Z$ and$\{Z_{i}\}_{i=1}^{m}$

This statement is meaningful. The original proof ([5]) of Theorem 1.1 was unsuccessful tocover the infinite dimensional setting. Concerning inequality (1.3), the original statement
is in the framework of the spectral order in a semi-finite von Neumann algebra. It is also
by adding a $rI$ term in the RHS with $r>0$
possible to give a version for operators on
arbitrarily small, see [6] pp. 11-15. By arguing as in [2] we then obtain:$\mathcal{H}$Let $A,$ $B\geq 0$ on$\mathcal{H}$then,and let $r>0$ . If$f$is a monotone concave function on $[0, \infty),$ $f(O)\geq 0$ ,$f(A+B)\leq Uf(A)U^{*}+Vf(B)V^{*}+rI$

for some unitart,es$U,$$V$.

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315 (1999) 771-78O.
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[10] F. HANSENAND G. K. PEDERSEN, Jensen s inequality for operators and Lowner s theorem, Math.
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[11] T. KOSEM, Inequalities between $\Vert f(A+B)$ I and 1 $f(A)+f(B)\Vert$ , Linear Alg. Appl., 418 (2006)

153-160.

[12] S. JU. ROTFEL’D, The singular values of a sum of completely continuous operators, Topics inMathematical Physics, Consultants Bureau, Vol. 3 (1969) 73-78.
[13] B. SIMON, Trace Ideals and Their Applications, LMS lecture note 35, Cambridge Univ. Press,Cambridge, 1979.
[14] M. UCHIYAMA, Subadditivity of eigenvalue sums, Proc. Amer. Math. Soc., 134 (2006) 1405-1412.

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JEAN-CHRISTOPHE BOURIN, DEPARTEMENT DE MATH\’EMATIQUES UNIVERSIT\’E DE $FRANCHE-CoMT\acute{E}$
ROUTE DE GRAY, 95030 BESAN $qoN$ , FRANCE
E-mail address: ean-chr ist ophe. bour inQuniv-f omt .fr
$j$

$c$

$e$

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CONCAVE FUNCTIONS OF POSITIVE OPERATORS, SUMS AND CONGRUENCES

EUN-YOUNG LEE. DEPARTMENT OF MATHEMATICS KYUNGPOOK NATIONAL UNIVERSITY DAEGU

702-701, KOREA

E-mail address. $eylee89\emptyset knu$ . ac. kr