数理解析研究所講究録
979 巻 1997 年 22-28

22Characterizations of operator convex functions of several
variables
Frank Hansen

Introduction

1

be a real function of variables defined on the product of:Let$x=(x_{1},\ldots,x_{k})$ be a tuple of selfadjoint matrices of ordersuchintervals, and let. We say that such a tuple
that the eigenvalues of are contained in for each
is in the domain of and define $f(x)=f(x_{1}, \ldots, x_{k})$ to be the matrix of order
we consider the possibly degenerate
constructed in the following way. For each
spectral resolution
$k$

$k$

$I_{1}\cross\cdots\cross I_{k}arrow \mathrm{R}$$f$$n_{1},$$\dot{i}=1,$$I_{i}$$x_{i}$$n_{k}$$\ldots,$$k$$\ldots,$$f$$n_{1}\cdots n_{k}$$\dot{i}=1,$$k$$\ldots,$$x_{i}= \sum_{m?1=}^{n}\lambda_{m_{i}}(_{i}i)e_{m_{i}m_{i}}^{i}$where$\{e_{s_{i}u_{t}}\}_{s}n.iu_{i}=1$

is the corresponding system of matrix units and let the formula.. .

$f(x_{1}, \ldots, xn)=\sum n_{1}$$m_{1}=1$$\sum n_{k}f(\lambda_{m_{1}}(1), \ldots, \lambda_{m_{k}}(k))e_{m11}\otimes\cdots\otimes e1mm_{k}mkk$$m_{k}=1$

define the functional calculus. If can be written as a product of functions $f=f_{1}\cdots f_{k}$coordinate, then $f(x_{1}, \ldots, x_{k})=f_{1}(X_{1})\otimes\cdots\otimes f_{k}(X_{k})$ .
where is a function only of the
The given definition is readily extended to bounded normal operators on a Hilbert space,cf. [7].
$k$$f$$\dot{i}’ \mathrm{t}\mathrm{h}$$f_{i}$

The above function
$(*)$

$f$

of

$k$

real variables is said to be matrix convex of order$(n_{1}, \ldots, n_{k})$, if$f(\lambda x_{1}+(1-\lambda)y_{1}, \ldots, \lambda Xk+(1-\lambda)y_{k})\leq\lambda f(X_{1}, \ldots, X_{k})+(1-\lambda)f(y1, \ldots, y_{k})$

and
such
for every $\lambda\in[0,1]$ and all tuples of selfadjoint matrices.
and their eigenvalues are contained in for $i=1,$
that the orders of. and are.
contained
is
in
the
of
The definition is meaningful since also the spectrum. It is clear that the pointwise limit of a sequence of matrix
interval for each $i=1,$. If
isis again matrix convex of orderconvex functions of order, then it is also matrix convex of any ordermatrix convex of order. If
is matrix convex of all orders, then we say thatfor $\dot{i}=1,$such thatare open intervals, then it is enough to assume thatis operator convex. If
is mid-point matrix convex of arbitrary order. This follows because such a function is real
analytic and hence continuous, cf. the discussion in the introduction of [4].
$(x_{1}, \ldots, x_{k})$$x_{i}$$n_{i}$$y_{i}$$(y_{1}, \ldots, y_{k})$$k$$I_{i}$$\ldots,$$\lambda x_{i}+(1-\lambda)y_{?}$$k$$I_{i}$$\ldots,$$(n_{1}, \ldots, n_{k})$$(n_{1}, \ldots, n_{k})$$(n_{1}, \ldots, n_{k})$$k$$n_{i}’\leq n_{i}$$f$$\ldots,$$I_{1},$$I_{k}$$\ldots,$$f$$(n_{1}’, \ldots , n_{k}’)$

$f$

$f$

23

2 JENSEN’S OPERATOR INEQUALITY

Jensen’s operator inequality

2

The following theorem for functions of one variable were proved in [6].
Theorem 2.1 If, the folllowing$\infty)$

(1)

$f$

$f$

is a continuous, real function on the half-open interval [are equiavalent:$0,$$\alpha$[ (with$\alpha\leq$$cond?\text{ノ}ti_{onS}$,

is operator convex and$f(\mathrm{O})\leq 0$(2) $f(a^{*}[0,\alpha[x.a)\leq a^{*}f(x)a$ for alll$a$.

with $||a||\leq 1$ and every $sel,f$-adjoint(3) $f$ (p.xp) $\leq pf(x)p$ for every projection$p$and every selfadjoint

$x$

$x$

with spectrum, $in$with spectrum in $[0,$$\alpha[$.

Aujla [1] extended the previous result in 1993 and essentially proved the following theorem:
Theorem 2.2 If[ (with[$0,$$\alpha[\cross[0,$

(1)

$f$

$\beta$$f$$\alpha,$.

is a real continuous function of two variables defined on the domaine), the following conditions are equivallent:.
$\beta\leq\infty$

is separately operator convex, and $f(t, 0)\leq 0$ and $f(0, s)\leq 0$ for all $(t, s)\in$$[0,$ $\alpha[\cross[\mathrm{o},$ $\beta[$.

(2) $f(a^{*}Xa, a*ya)\leq(a^{*}\otimes a)f(x, y)(a\otimes a)$ for all with $||a||\leq 1$ and all selfadjoint $x,$
$w?,th$ spectra contained in [[ respectively.
[ and [
$a$

$0,$

$0,$

$\alpha$(3) $f(pxp,pyp)\leq(p\otimes p)f(x, y)(p\otimes p)$ for every projection[ respectively.
[ and [
spectra contained in [
$0,$

$y$

$\beta$$p$

and all selfadjoint$x,$ $y$with$0,$ $\beta$$\alpha$The above operator inequality is equivalent to$f(a^{*}Xa, b*yb)\leq(a^{*}\otimes b^{*})f(x, y)(a\otimes b)$

for arbitrary contractions and , but this generalization is not essential. The class of
separately operator convex functions is evidently not of much importance, but Aujla’s result
paved the road for further progress. The next result [4] followed in 1996.
$a$

is a real continuous function of two variables defined on the domaine$\beta[(w?,th\alpha, \beta\leq\infty)$ , the folllowing $condit?,ons$ are equivalent:

Theorem 2.3 If[$0,$$\alpha[\cross[0,$

(1)

$f$

$b$

$f$

$\leq$

is operator convex, and $f(t, 0)\leq 0$ and$f(\mathrm{O}, s)\leq 0$

for all $(t, s)\in[0,$$\alpha[\cross[0,$$\beta[$.

(2) The operator $?,nequality$

is valid for all selfadjoint operators and $yw?,th$ spectra in [and all pairs of operators $(a, b)$ such that $aa^{*}+bb^{*}=1$ and
$x$

$0,$

$b$

[ respectively,[ and [
is normal.

$\alpha$$0,$ $\beta$

24

3 GENERALIZED HESSIAN MATRICES(3) The operator inequality$\leq$

is valid for all selfadjoint operatorsand every ortogonal projection .

$x$

and

$y$

with spectra in [$0,$$\alpha$[ and $[0,$ $\beta[respectivel,y_{f}$$p$The characterization of operator convexity by a suitable generalization of Jensen’s operator
inequality has recently been extended to functions of several variables by H. Araki and theauthor.

3

Generalized Hessian matrices

The notion of partial divided differences plays an important role in differential analysis of
matrix and operator convexity. The first divided difference of a differentible function of one
variable goes back to Newton. It is defined as$\frac{f(\lambda)-f(\mu)}{\lambda-\mu}$for$\lambda$$f’(\lambda)$

for$\lambda$$[\lambda\mu]=\{$and it is a symmetric function of the two arguments. If
second divided difference
is defined as$\neq\mu$$=\mu$$f$is twice differentiable, then the$[\lambda\mu\zeta]$$\frac{[\lambda\mu]-[\mu\zeta]}{\lambda-\zeta}$for$\lambda$$\neq\zeta$

for$\lambda$$=\zeta$$[\lambda\mu\zeta]=\{$$\frac{\partial}{\partial\lambda}[\lambda\mu]$

and it is a symmetric function of the three arguments, cf. [2] for a more systematic introduction to divided differences for functions of one variable.

If $f$ is a real function defined on the productof two open intervals with continuous
partial derivatives up to the second order, then we can consider the divided differences
andwhich are just the previously defined divided differences for the function of one
variable obtained by fixing the second variable to . We define the divided differencesandsimilarly. There are, however, also mixed second derivatives defined as$I_{1}\cross I_{2}$$[\lambda\mu|\xi]$$[\lambda\mu\zeta|\xi]$$\xi$$[\xi|\lambda\mu]$$[\xi|\lambda\mu\zeta]$$\frac{[\lambda|\zeta\xi]-[\mu|\zeta\xi]}{\lambda-\mu}$for$\lambda$

for$\lambda$$\neq\mu$$[\lambda\mu|\zeta\xi]=\{$$\frac{\partial}{\partial\lambda}[\lambda|\zeta\xi]$$=\mu$.

25

3 GENERALIZED HESSIAN MATRICES
We could have defined the mixed derivatives by dividing to the right instead of dividing to
the left, but this gives the same result. Finally, if $f$ is a real function defined on the productof open intervals with continuous partial derivatives up to the second order,
all but one or two of
divided differences that appear bythen we consider thethe coordinates of . They are labeled as$k$$I_{1}\cross\cdots\cross I_{k}$$\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{o}\dot{\mathrm{n}}\mathrm{d}$$k$$\mathrm{f}\mathrm{i}\mathrm{x}\mathrm{i}\dot{\mathrm{n}}\mathrm{g}$$f$$[\lambda_{1}|-\cdot\cdot|\mu 1\mu_{2}\mu 3|\cdots|\lambda_{k}]^{i}$where the superscript indicates that the partial divided difference of the second order isandcoordinate and all other coordinates are fixed at the values
taken at the
,
or as
$i$

$\dot{i}’ \mathrm{t}\mathrm{h}$$\lambda_{i-1}$$\lambda_{1},$$\ldots,$$\lambda_{i+1}\ldots$$\lambda_{k}$$[\lambda_{1}|\cdots|\mu 1\mu_{2}|\cdots|\xi_{1}\xi 2|\cdots|\lambda_{k}]^{ij}$where the superscripts $ij$ indicate that the mixed partial divided difference of the second
order is taken at the distinctly different coordinates and and all other coordinates are. The notation does not imply
andfixed at the values
any particular order of the coordinates which can be chosen from the full range 1, . . . , . The
following definition were introduced in [5]:..
$j$$\dot{i}$$\lambda_{i-1},$$\lambda_{1},$$\ldots,$$\lambda_{i+}1,$$\ldots,$$\lambda_{j+1},$$\lambda j-1$$\lambda_{k}$$\ldots,$$k$$\cdot$be a real function of variables defined on the
Definition 3.1 Let :$\dot{i}nterval,Sw?,th$,continuous partial derivatives up to the second order. $We$
product of open
for to be an element
define a data set A of order.
write it in th ノ e form
$f$

$k$

$I_{1}\cross\cdots\cross I_{k}arrow \mathrm{R}$$k$$(n_{1}, \ldots, n_{k})$$f$$\Lambda\in I_{1}^{n_{1}}\mathrm{x}\cdots\cross I_{k}^{n_{k}}.’$$a..n.\cdot dw..e$$usuall\prime y$$k_{\nu_{}}$$(*)$$\dot{i}=1,$$\Lambda=\{\lambda m_{i}(\dot{i})\}mi=1,\ldots,n_{i}$$k$$\ldots,$.

To a given data set A we $assoC?,ate$ so-callled generalized $Hess?,an$ matrices. First we defineato eachof natural numbers $(m_{1}, \ldots, m_{k})\leq(n_{1}, \ldots, n_{k})$ and to any $u=1,$
way:
in thematrix denoted $H_{su}(m_{1}, \ldots, m_{k})$ of order
$k$

$s,$

$tup^{1_{\text{ノ}}}e$$\ldots,$$f_{oll_{\mathit{0}}w}?,ng$$n_{u}\cross n_{s}$1. If $s\neq u$ , then we set$H_{su}(m_{1}, \ldots, m_{k})=$$([\lambda_{m_{1}}(1)|\cdots|\lambda_{m_{s}}(s)\lambda j(s)|\cdots|\lambda_{p}(u)\lambda m_{u}(u)|\cdots|\lambda mk(k)]Su)_{p}=1,\ldots,nu;j=1,\ldots,n_{S}$2. If $s=u$ , then we set$H_{SS}(m_{1,\ldots,k}m)=2([\lambda m_{1}(1)|\cdots|\lambda(S)\lambda_{p}(_{S})\lambda_{j(S})m_{s}|\cdots|\lambda_{m}(kk)]^{S)_{p,j=1,\ldots,n_{S}}}$We then define th, generallized Hessian matrix as the block matrix$e$$H(m_{1}, \ldots, m_{k})=(H_{su}(m1, , .

.

, m_{k}))_{u,s=1,\ldots,k}$which is quadrati and symmetric and of order $n_{1}+\cdots+n_{k}$ .
$c$and there
If $n_{i}=1$ for $i=1,$
then the data set $(*)$ reduces to numbers$H$matrix with theis only one (generalized) Hessian matrix . The submatrix
is a 1. Therefore $H$ canas matrix element for $u=1,$
partial derivative
be identified with the usual Hessian matrix associated with a function of variables.
$k$

$k$

$\lambda(1),$$\ldots,$$H_{su}$$f_{S}’’u(\lambda(1), \ldots, \lambda(k))$$s,$$\ldots,$$\cross 1$$k$$\ldots,$$k$$\lambda(k)$26

4 DIFFERENTIAL CHARACTERIZATION OF MATRIX CONVEXITY

4

Differential characterization of matrix convexity

The functional calculus $(x_{1}, \ldots, x_{k})arrow f(x_{1}, \ldots, x_{k})$ for functions of several variables defines
a mapping from (a subset of) the direct sum $B(H_{1})\oplus\cdots\oplus B(H_{k})$ to the tensor product$B(H_{1})\otimes\cdots\otimes B(H_{k})$ . The mapping is twice Fr\’echet differentiable, if
has continuous partial
derivatives of order $p>2+k/2$ , cf. [5, Corollary 2.12]. For $k\leq 2$ there are sharper results. Vasudeva, and it may well be that $p=2$ is a both necessary and. Brown andbysufficient condition for general . The following result is of a classical nature and can bederived from [3].
$f$$\mathrm{H}.\mathrm{L}$$\mathrm{A}.\mathrm{L}$

$k$

. .
have
Theorem 4.1 Let the Hilbert spaces
Fr\’ech,etdifferentiable, thenfunctional calculus $m,app?,ng$ is twice
only
and
if
if
$H_{1},$$(n_{1}.’...\cdot.

$fi_{\text{ノ}}n\dot{i}ted\dot{i}m,enS?\prime onsn_{1},$$H_{k}$

$,$

$,$

$f$

is

$matr?,X$

.

, ,

,

$n_{k}$. If theconvex of order, n_{k})$$d^{2}f(x_{1}, \ldots, X_{k})(h, h)\geq 0$

for any tuple$h=(h^{1}, \ldots, h^{k})$of $selfadjo?,ntm,atriceS$ on$H_{1},$$\ldots,$$H_{k}$.

The above result is of great import in conjunction with the following structure theorem forthe second Fr\’echet differential, cf. [5].
with $p>2+k/2$ where , , .., are open intervals
Theorem 4.2 Let
be selfadjoint matrices of orders
and let $x=$
in the domain of .
. .. ,
The $expeCtat?,on$ value of the second Fr\’echet
in a vector
isgiven by$f\in C^{p}(I_{1}\cross\cdots\cross I_{k})$$(x_{1},$$I_{k}$$I_{1}$$(n_{1}, \ldots, n_{k})$$x_{k})$$f$$\varphi\in H_{1}\otimes\cdots\otimes H_{k}$$di_{\text{ノ}}fferent’,al$$(d^{2}f(x)(h, h) \varphi|\varphi)=\sum_{m_{1^{=}}1m_{k}}^{n_{1}}\cdots\sum(H(m_{1},$$m_{k}nk=1\mathrm{I}\Phi^{h}(m_{1},\ldots , m_{k})|\Phi^{h}(m1, \ldots, mk)\mathrm{I}$$\ldots,$

where $H(m_{1}, \ldots, m_{k})$ are the generallized Hessian matrices $assoc\dot{?,}ated$ with,$wh?,l,e$ the vectors
values of$f$and the eigen-$(x_{1}, \ldots, x_{k}),$$\Phi^{h}(m_{1}, \ldots, m_{k})=$$m_{i}=1,$ $\ldots,$$n_{i}$

for$i=1,$$k$$\ldots,$

are $g?,ven$ by$\Phi_{s}^{h}(m_{1}, \ldots, m_{k})_{j_{s}}=h_{m_{s}jS}^{S}\varphi(m_{1}, \ldots, m-1,j_{s}S’ 1, \ldots,k)m_{S+}m$for $j_{s}=1,$$\ldots,$$n_{s}$

and $s=1,$$k$$\ldots,$.

, one canWe immediately realize that even without calculating the vectors,
providedconclude that is matrix convex of order
all of the generalized Hessian$\Phi^{h}(m_{1}, \ldots, m_{k})$$f$$(n_{1}, \ldots, n_{k})$27

REFERENCES
matrices associated with and any data set
This can for example be done for the functions$\Lambda\in I_{1}^{n_{1}}\mathrm{x}\cdots\cross I_{k}^{n_{k}}$$f$$f(t_{1}, \ldots, t_{k})=\prod_{i=1}^{k}\frac{1}{1-\mu_{i}t_{i}}$$t_{1},$$\ldots,$are positive semi-definite.

$t_{k}\in]-1,1[$$\mu_{k}\in[-1,1]$ . It is calculated in [5] that the generalized Hessian matrices for
wherethese functions are of the form$\mu_{1},$$\ldots,$$a(2)^{t}\cdot a(1)$$2a(2)^{t}\cdot$a(1)^{t}..\cdot\cdot a(2)$a(2)$.

$H(m_{1}, \ldots, m_{k})=f(\lambda m_{1}(1), \ldots, \lambda m_{k}(k))$$2a(k)t.a(k)a(2)^{t}.\cdot.a(k)a(1)t.\cdot a(k))$$/\backslash 2a(1)^{t}.\cdot.\cdot a(1)$$..$$a(k)^{t}\cdot a(1)$.

$a(k)^{t}\cdot a(2)$where the vectors$a(i)=\mu_{i}(fi(\lambda_{1}(i)),$$fi(\lambda n_{i}(i)))\in \mathrm{R}^{n_{2}}$$\ldots,$

for $i=1,$ $\ldots} k$ . The generalized Hessian matrices are bounded from below by$f(\lambda_{m_{1}}(1), \ldots, \lambda_{m_{k}}(k))$$=$$f(\lambda_{m_{1}}(1), \ldots, \lambda_{m_{k}}(k))(a(1)$$a(k))$$a(k))^{t}(a(1)$which are positive semi-definite matrices.
Corollary 4.3 Let be a non-negative Borel measure on the cube -1,
The function$a_{1,\ldots,k}a$ be real, numbers.
let
$[$

$\nu$

$1]^{k}$for$k\in \mathrm{N}$and$a_{0},$$f(t_{1}, \ldots, t_{k})=a_{01}+at_{1}+\cdots+a_{kk}t+\int^{1}-1\ldots\int_{-1}1\prod_{=i1}^{k}\frac{1}{1-\mu_{i}t_{i}}d\nu(\mu_{1}, \ldots, \mu_{k})$is operator convex on the open $cube$ ] $-1,1[^{k}$ .

References[1] J.S. Aujla. Matrix convexity of functions of two variables. Linear Algebra and Its Applications, 194:149-160, 1993.
[2] W. Donoghue. Monotone matrix functions and analytic continuation. Springer, Berlin,
Heidelberg, New York, 1974.

28

REFERENCES[3] T.M. Flett. Differential Analysis. Cambridge University Press, Cambridge, 1980.
[4] F. Hansen. Jensen’s operator inequality for functions of two variables. to appear in Proc.
Amer. Math. Soc, 1996.
[5] F. Hansen. Operator convex functions of several variables. RIMS-lll9, 1996.
[6] F. Hansen and G.K. Pedersen. Jensen’s inequality for operators and L\"owner’s theorem.
Math. Ann., 258:229-241, 1982.
[7] A. Kor\’anyi. On some classes of analytic functions of several variables. Trans Amer.
Math. Soc., 101:520-554, 1961.