数理解析研究所講究録
第 1996 巻 2016 年 45-53

45

RELATIVE OPERATOR ENTROPY AND KARCHER MEAN
JUN ICHI FUJII1 AND YUKI SE02

1.

$IN^{r}I’$

RODUCTION

Lawson and Lim [10] showed that the Karcher equation for positive invertible operators, $X$ on a Hilbert space and a weight:$A_{j}(j=1,2, \ldots, n)$(KE)$\{\omega_{j}\}(\omega_{j}\geq 0, \sum_{j}\omega_{j}=1)$$0= \sum_{j=\lambda}^{n}\omega_{j}\log(X^{-\frac{1}{2}}A_{j}X^{-\frac{1}{2}})$has a unique positive invertible solution$X=G_{K}(\omega_{j};A_{j})=G_{K}(\omega;\mathbb{A})$for. It is called the (weighted -variable) Karcher
mean. This definition depends on the invertibility of operators: Their theory depends
on the Thompson metric $d(A, B)=\Vert\log(A^{-1/2}BA^{-1/2}\Vert$ and the power operator mean
corresponding to the power function: The Karcher mean of
positive invertible operators coincides with the strong operator limit of the power means
as $tarrow 0$ :.
$\omega=$$(w_{1}, .

.

.

, \omega_{n})an(i\mathbb{A}=(A_{1}, \ldots, A_{n})$$n$$f_{m_{r,t}}(x)=(1-t+tx^{r})^{\frac{1}{\gamma}}$$P_{t}(\omega;A)$$G_{K}( \omega;A\rangle=s-\lim_{tarrow 0}P_{t}(\omega_{)}\cdot A)$

Thus it needs substantially the invertibility of positive operators.
In this note, we extend it to a mean for (non-invertible) positive operators by virtueof the relative operator entropy. So, wee observe the properties for the relative operator
entropy with the existence conditions which are closely related the kernels and ranges for
operators. Defining the Karcher mean for nbn-invertible operators as the strong-operator
limit, we discuss their properties and kernels. Then we see that the original Kubo-Andogeometric mean satisfies the Karcher equation with the kernel condition. Also we verify
that the Karcher mean for operators with the closed ranges is a unique solution of thisequation.

2. RELATIVE OPERATOR ENTOROPYFirst we review the relative operator entropy $S(A|B)$ for positive (bounded linear)
operators $A,$
on a Hilbert space, see [5, 6, 7, 2, 3, 4].
Nakamura and Umegaki [11] extended the notion of the entropy formulated by von
Neumann and gave the entorpy by $-A\log$ for a positive operator on a Hilbert space.
Also, Umegaki [13] introduced the relative entorpy as a noncommutative version of the
Kullback-Leibler entorpy, which is given by the trace of Alog $A-A\log B$ for positive
operators $A,$ $B$ aﬃliated with a semifinite von Neumann algebra. In [5], Fujii and Kamei
$B$

$A$

$A$

2000 Mathematics Subject Classification. Primary $47A64$ ; Secondary $47A63,$ $47Aii8,$ $94A17.$
Key words and phrases. Operator mean, relative operator entropy, Karcher equation, Karcher mean.

46
introduced the relative operator entorpy which is a relative version of the operator entorpy
defined by Nakamura-Umegaki: Let and be positive operators on a Hilbert space $H.$
If is invertible, then the relative operator entorpy is defined by
$A$

$B$

$B$

$S(A|B)=B^{\frac{1}{2}}\eta(B^{-\frac{1}{2}}AB^{-\frac{1}{2}})B^{\frac{1}{2}}$where$\eta$

is the entropy function:$\eta(x)=$In addition, if$\mathcal{A}$-xlog

if $x>0,$$x$$\eta(0)=0.$

is invertible, then the relative operator entropy is rewrited as follows:$S(A|B)=A^{\frac{1}{2}}\log(A^{-\frac{\lambda}{2}}BA^{-\frac{1}{2}})A^{\frac{1}{2}}.$In the case, since$\log t$is operator monotone, $S(A|B)$ has the right monotonicity:$B\leq C$

Moreover, by using the fact that
[12]:

implies$S(A|B)\leq S(A|C)$ .

$\lim_{tarrow 0}\frac{x^{\ell}-1}{t}=\log t$, it is constructed by Uhlmann’s way$S(A|B)= s-\lim_{tarrow 0}\frac{A\#_{t}B-A}{t},$where the geometric operator mean $A\#_{t}B$ is defined by$A\#_{t}B=A^{1/2}(A^{-1/2}BA^{-1/2})^{t}A^{1/2}.$

Next, letbe non-invertible. For eachandrelative operator entropy $S(A|B+\epsilon)$ exists and
$A$

$B$

$\epsilon>0$, since$B+\epsilon$

is invertible, the$S(A|B+ \epsilon)=s-\lim_{\delta\downarrow 0}S(A+\delta|B+\epsilon)$in the strong operator topology. Since for each $\delta>0,$ $S(A+\delta|B+\epsilon)$ has the right-term, we havemonotone decreasing property as$S(A+\delta|B+\epsilon)\geq S(A+\delta|B+\epsilon’)$for $g>e’>0$$\epsilon\downarrow 0$

and so $S(A|B+\epsilon)\geq S(A|B+\epsilon’)$ as, we can define$\delta\downarrow 0$. Therefore, in the case of non-invertible

$A$

and

$B$

$S(A|B)= s-\lim_{\epsilon\downarrow 0}S(A|B+\epsilon)$

if the strong-operator limit exists as a bounded operator. In fact, by the right-termmonotonicity, the lower boundedness for $\{S(A|B+\epsilon)|\epsilon>0\}$ guarantees the strongoperator limit. But, in general, $S(A|B)$ does not always exist. For example, $S(A|B)$ does
not exists for invertible and non-invertible $B.$
Here, we state a existence condition of the relative operator entropy:$A$

Lemma 2.1. Let $A,$ $B$ be positive operators. The relative operator entropy $S(A|B)$ exists
if and onty if $E(\alpha)=(-1-\log\alpha)A+\alpha B$ is bounded below for all $a>0.$

Next, we consider another equivalent definition of Uhlmann’s type. For this, based on, it follows thatis monotone-decreasing as
the fact thatasso that another equivalent definition of Uhlmann’s type is the derivative one for the pathof geometric means $A\#_{t}B$ for $t\in(O, 1$ ]:$\frac{x^{\ell}-1}{t}\searrow\log t$$t\downarrow 0$$\frac{A\# eB-\mathcal{A}}{t}$$S(A|B)= s-\lim_{t\downarrow 0}\frac{A\#_{t}B-A}{t}$$t\downarrow 0,$

47
if the strong-operator limit exists. In fact, put$X= s-\lim_{t\downarrow 0}\frac{A\# tB-A}{t}$for each$\frac{A\#_{t}B-A}{t}\leq\frac{(A+\epsilon)\#_{t}(B+\epsilon)-(A+\epsilon)}{t}$we have. On the other hand, since$X \leq s-\lim_{t\downarrow 0}\frac{(A+\epsilon)\# t(B+\epsilon)-(A+\epsilon)}{t}=S(A+\mathcal{E}j|B+\epsilon)$as$t\downarrow 0$

and since$\epsilon:>0,$

and so $X\leq S(A|B)$ as$\epsilon\downarrow 0$

for each$\frac{(A+\epsilon)\#_{t}(B+\epsilon)-(A+\epsilon)}{t}\geq S(A+\epsilon|B+\epsilon)$$\epsilon>0$

and $t\in(O, 1$ ],we haveasfor $b>0$ and so $X\geq S(A|B)$ . Therefore, $X=S(A|B)$
and we have another equivalent definition of Uhlmann’s type for non-invertible case.
If and are commuting and $S(A|B)$ is defined, then$\frac{A\#\prime B-A}{t}\geq S(A|B)$$A$$\epsilon\downarrow 0$$B$$S(A|B)=A\log B-A\log A,$

in particular, $S(O|B)=0$ for all $B\geq 0$ . In fact, since $S(A|B)$ exists, by Lemma 2.1, there
exists a scalarsuch that for all $\alpha>0$$c\in \mathbb{R}$$c\leq(-1-\log\alpha)A+\alpha B$

for each$\leq(-1-\log a)A+\alpha(B+\epsilon)$$\epsilon>0$

and so$c\leq(B+\epsilon)(-(B+\epsilon)^{-1}A\log(B+\epsilon)^{-1}A)$$=-A\log A+A\mathfrak{i}og(B+\epsilon)$.

Therefore, since Alog $B-A\log A$ exists, we have A $\log B-A\log A=S(A|B)$ .
$A$

Under the existence, we have the following properties of $S(A|B)$ for positive operators
and $B$ by those for operator means:

Lemma 2.2. Under the existence, the following properties like operator means hold:(1) right monotonicity:, then $S(A|B)\leq S(A|B’)$ .
If(2) transformer inequality: $T^{*}S(A|B)T\leq S(T^{*}AT|T^{*}BT)$ for all(the equality holds for invertible ).
$\Phi(S(A|B))\leq S(\Phi(A)|\Phi(B))$(2’) informational monotonicity:
for all normal positive linear maps
$S(A_{1}|B_{1})+(A_{2}|B_{2})\leq S(A_{1}+A_{2}|B_{1}+B_{2})$ .
(3) sub-additivity:$(3’)$ joint concavity:$B\leq B^{l}$

$T$

$T$

$\Phi.$$(1-t)S(A_{1}|B_{1})+tS(A_{2}|B_{2})\leq S((I-t)A_{1}+tA_{2}|(1-t)B_{1}+tB_{2}\rangle$

for $ailt\in[0,1].$$S(A|B)\leq B-A.$(4) upper bound:$kerS(A|B)\supset kerA.$(5) kernel inclusion:$S(\oplus_{k}A_{k}|\oplus_{k}B_{k})=\oplus_{k}S(A_{k}|B_{k})$ .
(6) orthogonality:(7) aﬃne parametrization: $S(A|A\#_{t}B)=tS(A|B)$ for all $t\in[0$ , 1$].$

Here we recall the equality condition for the transformer inequality (2) of Lemma 2.2:
Lemma 2.3. Let
andequalitytransformer
$A$

$B$

be positive operators. Under the existence of $S(A|B)$ , the$T^{*}S(A|\mathcal{B})T=S(T^{*}AT|T^{*}BT)$holds for an operator $TMthkerT^{*}\subseteq$ kerA$\cap$kerB.

48
We have the following relations around the existence condition for the relative operator
entropy:hold in the following conditions
Theorem 2.4. The implications (1)$B\geq0$$A,$and each converse does not always hold.
for a pair ofo $rr$ ange i $nc1usion:\exists\alpha>0;A\leq\alpha B,i.e.,$existence condition.$\cdot S(A|B)$ exists a$saboundd\acute{d}$$\Rightarrow(2)\Rightarrow(3)\Rightarrow(4)$$(2)(1)$

operatormajorization’$ranA^{\frac{1}{2}}i.e.,\subset ranB^{\frac{1}{2}}.$$\inf_{\alpha>0}[\frac{1}{\alpha}B-A+(\log\alpha)A]>-\infty.$

(3)

$B$

-absolute continuity:(4) kernel inclusion:$A=[B]A(=A^{\frac{1}{2}}P_{M}A^{\frac{1}{2}}= \lim_{t\downarrow 0}A\#_{t}B)$.

$kerA\supset kerB.$

Remark 2.1. If both ranges of and are closed, in particular, for the case of matrices,
the above conditions in Theorem 2.4 are all equivalent since the relation ran
holds for all positive operators
$A$

$B$

$A^{\frac{1}{2}}=\overline{ranA}=$$A.$$(kerA)^{\perp}$

For invertible positive operators and , it is easy to see that the positivity (resp.
negativity) of $S(A|B)$ is equivalent to $B\geq A$ (resp. $A\geq B$ ) and hence $S(A|B)=0$ if
and only if $A=B$ . Second we discuss non-invertble case:
$A$

$B$

and $B$ be positive operators. Suppose $S(A|B)$ exists. $S(A|B)\geq 0$
Theorem 2.5. Let$B\geqA.$
If $ker$ is trivial, then $S(A|B)\leq 0$ if and only if $A\geq B.$
if and only if
Consequently, for with the trivial kernel, $S(A|B)=0$ if and only if $A=B.$
$A$

$A$

$A$

3. KARCHER MEAN FOR NON-INVERTIBLE POSITIVE OPERATORS
the Karcher
For non-invertible positive operators $A_{j}(j=1,2, \ldots, n)$ , for eachguarantees the strongexists and the monotonicity ofmean$\epsilon>0$$G_{K}$$X_{\epsilon}=G_{K}(\omega_{J}\prime;A_{j}+\epsilon)\geq 0$

operator limit:$X_{0}= s-\lim_{\epsilonarrow 0}X_{\epsilon}=s-\lim_{\epsilonarrow 0}G_{K}(\omega_{j};A_{j}+\epsilon)$.

Karcher meanand call it
Naturally we write $X_{0}=G_{K}(\omega_{j};A_{j})$ for non-invertible
again.
: The
synchronously toHere we extend the extremal means with a weight$H$
are defined byfor non-invertiblearithmetic mean and the harmonic one$\acute{t}he$$A_{j}$$G_{K}$$\{\omega_{j}\}$$A$$A_{j}$$A( \omega_{j};A_{j})=\sum_{j}\omega_{J}A_{j}$$H( \prime;)=s-\lim_{\epsilonarrow 0}H(\omega;A+\epsilon)=s-\lim_{\epsilonarrow 0}(\sum_{j}\omega\prime(A_{j}+\epsilon)^{-1})^{-1}.$

As for this construction of corresponding mean, we say $H$ is the adjoint of
Kubo-Ando theory.

$A$ ’

as in theand positive operators $A_{j}(j=1,2, \ldots, n)$ , the following
Theorem 3.1. For a weightproperties hold:.
(1) monotonicity: If
for all , then(2) transformer inequality,
for all(the equality holds for invertible ).
$\{\omega_{j}\}$$A_{j}\leq B_{j}$$\cdot$$j$$G_{K}(\omega_{j};A_{j})\leq G_{K}(\omega_{j};B_{j})$$T^{*}G_{K}(\omega_{j};A_{j})T\leq G_{K}(\omega_{j};\dot{T}^{*}A_{j}T)$

$T$

$T$

49
$(2^{f})$informational monotonicity:$\Phi(G_{K}(w_{j};A_{j}))\leq G_{K}(\omega_{j};\Phi(A_{j}\rangle)$

for all normal positive linear maps(3) sub-additivity.
(3’) joint concavity:$G_{K}(\omega_{j};A_{j})+G_{K}(\omega_{j};B_{j})\leq G_{K}(\omega_{j};A_{j}+B_{j})$$\cdot$$(1-t)G_{K}(\omega_{j};A_{j})+tG_{K}(\omega_{j};B_{j})\leq G_{K}(\omega_{j}, (1-t)A_{j}+tB_{j})$$\Phi.$.

for all $t\in[O$ , 1$].$(4) consistency with scalars:

If all$A_{j}$are commuting, then(5) self-adjointness:$G_{K}( \omega_{j};A_{j})=\prod_{j=1}^{n}A_{j}^{\omega_{J}}$with convention $A^{0}=I.$$G_{K}( \omega_{j};A_{j})=s-\lim_{\epsilon\downarrow 0}G_{K}(\omega_{j};\langle A_{j}+\epsilon)^{-1})^{-1}.$(6) joint homogeneity:$G_{K}( \omega_{j};c_{j}A_{j})=\prod_{J=1}^{n}c_{j}^{\omega_{j}}G_{K}(\omega_{j\rangle}A_{j}\rangle$

for non-negative numbers $c_{j}\geq 0(j=1,2, \ldots, n)$ .

(7) AGH weighted mean inequality:$H(\omega_{j};A_{j})\leq G_{K}(\omega_{j};A_{j})\leq A(\omega_{j};A_{j})$(8) orthogonality:.

$G_{K}( \omega_{j};\bigoplus_{tn}A_{j,m})=\bigoplus_{m}G_{K}(\omega_{j};A_{j_{\backslash }m})$ ,is nothing but $(n-1)$ -mean
Note that if
for some , then -mean
for all . Then
. So we call $G_{K}(w_{j};A_{j})$ the proper Karcher mean if
without
proper.
we also call the weight$k$$\omega_{k}=0$$G_{K}(\omega_{j};A_{j}\rangle$$n$$\omega_{k_{\rangle}}A_{k}$$\omega_{J}\prime>0$$j$$\{\omega_{j}\}$Lemma 3.2. For a proper $u/eight$ran$A( \omega_{j}; A_{j})^{\frac{1}{2}}=\bigvee_{j}$$\{\omega_{J}\grave{\}}$ran

and positive operators $A_{j}(j=1,2, \ldots, n)$ ,$A^{\frac{1}{j2}}$

and ran$H( \omega_{j}; A_{j})^{\frac{1}{2}}=\bigcap_{J^{J}}$ran$A^{\frac{1}{j2}}.$

By Lemma 3.2 and the ansformer inequaity (2) of Theorem 3.1, we have the following
kernel condition for the Karcher mean.
$t_{1}\cdot$

Theorem 3.3. Let$A_{j}$

be positive operators $forj=:$ , 2, . . . ,or a proper weight$F\fbox{Error::0x0000}$$n.$$\{\omega_{j}\},$$kerG_{K}(\omega_{j};A_{j})=\vee kerA_{j}j.$

For any projections and , it is known in [9] that $P\#_{t}B=P\wedge Q$ for $t\in[O$ , 1 . The
following result is an extension of 2-variable case:
$P$

$Q$

$]$

Corollary 3.4. For a proper weight$G_{K}(a)j;P_{j})= \bigwedge_{j}F_{j}$$\{\omega_{J}\hat{\}},$

for projections $P_{j}(j=1$ , 2, . . . ,$n\rangle.$

In this invertible case, note that (KE) is equivalent to a simple equation by the relative
operator entoropy$(**)$$0= \sum_{j=1}^{n}\omega_{i}S(X|A_{j})$,. But this equation always has a trivial
which also makes sense for non-invertible
solution $X=0$ since $S(0|A_{j})=0$ . For the case of Corollary 3.4, the entropy is $S(P|P_{j})=$$A_{j}$

50$P\log P_{j}-P\log P=0$ , and hence $P$ is indeed a solution of the equation $(**)$ .

But thisconsideration shows that each projection with $0\leq Q\leq P$ is a solution of $(**)$ .
Thereby, a reasonable extension of (KE) is the following EKE(Extended Karcher equation) with the kernel condition under the existence of each $S(X|A_{g})$ :
$Q$

(EKE)

$0= \sum_{j=1}^{n}\omega_{j}S(X|A_{j})$with$kerX=\omega_{j}>0\vee kerA_{j}.$Remark 3.1. If operators
are commuting for all , thenandis a solution of (EKE). But, if the kernel condition is removed, the following
example gives another solution $X$ even if $X$ commutes with all$X_{0}=G_{K}( \omega_{j};A_{j})=\prod_{j}A_{j}^{\omega_{j}}$$j$$A_{j}$$X_{0}$$A_{j}.$

Example 1. For diagonal matrices $A=$ diag $(a, b, c, 0)$ and $B=$ diag
$X_{1}=$ diag ( $0, b, 0,0)$ . Then, all matrices are commuting and hence$( \frac{1}{a}, b, 0, d)$, take$S(X_{1}|A)+S(X_{1}|B)=-2X_{1}\log X_{1}+X_{1}\log A+X_{1}\log B$

diag ( $0, -2b\log b, 0,0)+$ diag ( $0, b\log b, 0,0)+$ diag ( $0, b\log b, 0,0)=0.$

$=$

So

$X_{1}$

is a solution while $X_{1}\neq X_{0}=$ diag ( $1, b, 0,0)$ .

Remark 3.2. For the case of projections $A_{j}=P_{j}$ , the above is a unique solution of(EKE). In fact, suppose $Y$ is another solution. Then the kernel condition $P\leq P_{j}$ shows$PYP=Y$ and hence we have $YP_{j}=P_{j}Y=Y$ and $Y\log P_{j}=YP_{j}\log P_{j}=0$ . Therefore$P$$0= \sum_{j}\omega_{j}S(Y|P_{j})=\sum_{j}\omega_{j}(Y\log P_{j}-Y\log Y)$$= \sum_{j}\omega_{j}(-Y\log Y)=-Y\log Y,$

which implies that $Y$ must be a projection and consequently $Y=P$ by the kernel condi-

tion.

Moreover note that $S(A|B)$ does not always exist as a bounded self-adjoint operator$S(X_{0}|A_{j})$ indeed exists for the Karcher mean $X_{0}=$as in the preceding section. But$G_{K}(\omega_{j};A_{j})$:a weight. For the Karcher mean
Lemma 3.5. Let
be positive operators and$X_{0}=G_{K}(\omega_{j};A_{j})$ , each entropy $S(X_{0}|A_{j})$ exists$w_{j}>$
O. For $\omega_{k}>0$ , bounds areforexpressed by$\{\omega_{j}\}$$A_{j}$$- \frac{M_{k}}{\omega_{k}}\leq S(X_{0}|A_{k})\leq M_{k}$for$M_{k}= \max_{j\neq k}\Vert A_{j}\Vert+1.$

Thus far, we have not showed that our Karcher mean
for general positive operators. Here we can obtain only the inequality:$X_{0}= s-\lim_{\epsilonarrow 0}X_{\epsilon}$

Lemma 3.6. Let$A_{j}$

be positive operators for $j=1$ , 2, .. .,$0 \leq\sum_{j=1}^{n}\omega_{j}S(X_{0}|A_{j})$

with

$n$

and

$\{\omega_{j}\}$a weight. Then

$kerX_{0}=\omega_{j}>0\vee kerA_{j}.$However, in the case of $n=2$ , we have the following result:

satisfies (EKE)51
Theorem 3.7. For positive operators
mean $A\#_{f}B$ satisfies (EKE).

$A$

and $B$ and $t\in(O, 1)$ , the original geometric

Recall that a non-invertible positive operators has the closed range if and only if is
an isolated point in. Any positive semi-definite matrix has the closed range. Finallyin this section, we show that the Karcher mean for such operators is a unique solution of(EKE). To see this, we verify the following fact:
Lemma 3.8. If $A_{j}(j=1,2, \ldots, n)$ are positive opereators whose ranges are closed, then
so is
.
$A$

$0$

$\sigma(A)$$X_{0}=G_{K}(\omega_{j};A_{j})$So we have a unique solution of (EKE):
Theorem 3.9. If
the Karcher mean

$A_{j}$

are positive opereators whose ranges are closed for $j=1,$
is a unique solution of (EKE).

$2_{\}}\ldots$, $n,$$X_{0}=G_{K}(\omega_{j};A_{j})$Therefore, we have the following result in the matrix case:

Corollary 3.10. For positive semi-definite matrices
for $j=1$ , 2, .. . , and a weight, the Karcher mean $X=G_{K}(\omega_{j};A_{j})$ is the unique solution of (EKE):$n$$A_{j}$$\{w_{j}\}$$0= \sum_{j=1}^{n}w_{j}S(X|A_{J})$with$kerX=\omega_{j}>0\vee kerA_{j}.$4. POWER MEANS FOR NON-INVERTIBLE CASELet
be positive operators for $j=1$ , 2, . . . ,similarly to the Karcher mean

anda weight. For each, the power means$t\in(0,1]$ is the unique positive invertible solution of the power mean equation$A_{j}$$n$$\{\omega_{j}\}$$X_{\epsilon}=G_{K}(\omega_{j)}\cdot A_{j}+\epsilon)$$\epsilon>0,$$P_{t}(\omega_{i};A+\epsilon\rangle$

for$X= \sum_{j=1}^{n}\omega_{j}(X\#_{t}(A_{j}+\epsilon$

Then the Karcher mean for invertible case is the strong-operator limit of the powermeans:$s-\lim_{t\downarrow 0}P_{t}(\omega_{J}\prime;A_{j}+e:)=X_{\epsilon}.$The power means
are monotone decreasing for3.6 (4)] and lower bounded by the zero operator. Henceexists and$P_{t}(w_{j};A_{j}+\epsilon)$$\epsilon\downarrow 0$

by [10, Proposition$P_{t}( \omega_{j};A_{j})=\inf_{\epsilon>0}P_{t}(\omega_{j};A_{j}+\epsilon)$$P_{t}( \omega_{j};A_{j})=s-\lim_{\epsilon\downarrow 0}P_{t}(\omega_{j’}, A_{j}+\epsilon)$in the strong operator topology and so(1)$P_{t}(\omega_{i\rangle}A_{2})$is a solution of the power mean equation$X= \sum_{j}\omega_{j}(X\#_{t}A_{j})$for $t\in(0_{\}}1].$
Similarly to the Karcher equation for positive operators, the power mean equation (1)always has a trivial solution $X=0$ . Thereby, we consider the following EPE (Extended
Power mean equation) with the kernel condition:(EPE)$X= \sum_{j}\omega_{j}X\#_{t}A_{j}$

,

with

$kerX=\bigcap_{w_{j}>0}k_{\mathfrak{X}}A_{j}.$52be positive operators andfor $t\in(0,1]$ satisfy $(EPE)$ andTheorem 4.1. Let$P_{t}(\omega_{j};A_{J}\prime)$$A_{j}$$\{\omega_{j}\}$as$P_{t}(\omega_{j};A_{j})\searrow G_{K}(\omega_{j};A_{j})$Theorem $Y$ (Yamazaki [14]). Leta weight. Then

$A_{j}$

a weight. Then the power means$t\downarrow 0.$

and $X$ be positive operators for $j=1$ , 2, . . .,

$n$

and

$\{\omega_{j}\}$$\sum_{j}\omega_{j}S(X|A_{j})\geq 0$

Moreover, if$A_{j}$

implies$G_{K}(\omega_{j};A_{J}\prime)\geq X.$

and $X$ are invertible, then$\sum_{j}\omega_{j}S(X|A_{j})\leq 0$

implies$G_{K}(\omega_{j};A_{j})\leq X.$

The following result follows from the uniqueness of the power means and is an extensionof Theorem :$Y$

Theorem 4.2. Let
Then$A_{j}$

and $X$ be positive operators for $j=1$ , 2,

$X \leq\sum_{j}\omega_{j}X\#_{t}A_{j}$Moreover, if$A_{j}$

implies$\rangle n$$X\leq P_{t}(\omega_{j};A_{j})$.

$X\geq P_{t}(\omega_{j};A_{j})$.

and a weight$\{\omega_{j}\}.$

and $X$ are invertible, then$X \geq\sum_{j}\omega_{j}X\#_{t}A_{j}$implies

Remark 4.1. Theorem 4.2 is an extension of Theorem Y. Indeed, suppose
O. Then$\sum_{j}\omega_{J}\prime S(X|A_{j})$)$0 \leq\sum_{j}\omega_{j}S(X|A_{j}))\leq\frac{\sum_{j}\omega_{J}X\#_{t}A_{j}-X}{t}$and it follows from Theorem 4.2 thatfor all $0<t\leqq 1$ . Taking limit as, we have. Another part is obtained by this result.
$X\leq P_{t}(\omega_{j};A_{j})$$t\downarrow 0$$X\leq G_{K}(\omega_{j};A_{j})$Remark 4.2. Finally, considering the power means for negative parameters as the adjoint
in the sense of Kubo-Ando theory, for invertible case, we can show that the uniqueness of
the Karcher equation follows from the case of power means without the implicit function
theorem. See [8] for more detail,Though the properties for power means are supplied, the following general problems

are not still answered:
Conjecture. For non-invertible positive operators on a Hilbert space, the Karcher meansatisfies (EKE) and it is a unique solution of (EKE).

Conjecture 2. For non-invertible positive operators on a Hilbert space, each power mean
for $t\in(0,1] is a$ unique solution $of (EPE)$ .
$P_{t}(\omega_{j};A_{j})$$\geq$

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1 DEPARTMENT OF ART AND SCIENCES (INFORMATION SCIENCE), OSAKA KYOIKU UNIVERSITY,ASAHIGAOKA, KASHIWARA, OSAKA 5828582, JAPAN$E$-mail address: fuj iiQcc. osaka-kyoiku. ac. jp$2DEPARTMEN^{r}T$ OF MATHEMATICS EDUCATION, OSAKA KYOIKU UNIVERSITY, ASAHIGAOKA, KASHIWARA, OSAKA 582-8582, JAPAN-mail address: yukisQcc. osaka kyoiku. ac. jp$E$