数理解析研究所講究録 1458 巻 2005 年 89-96

89

REVERSE INEQUALITIES ASSOCIATED WITH TSALLIS RELATIVEOPERATOR ENTROPY VIA GENERALIZED KANTOROVICHCONSTANT
Takayuki Furuta 古田 孝之 (東京理大理)\S 1. Introduction
A capital letter means an operator on a Hilbert space $H$ . An operator $X$ is said to bestrictly positive (denoted by $X>0$ ) if $X$ is positive definite and invertible. For two strictly
is defined bypositive operators , and $p\in[0, 1]$ , -power mean
$A$$B$$A\#_{\mathrm{P}}B$$p$$A\# pB$$=A^{\frac{1}{2}}(A^{\frac{-1}{2}}BA^{\frac{-1}{2}})^{p}A^{\frac{1}{2}}$and we remark that $A\# pB$ $=A^{1-p}B^{p}$ if$A$

commutes with$B$.

Very recently, Tsallis relative operator entropy $T_{p}(A|B)$ in Yanagi-Kuriyama- uruichi[17] is defined by(1.1)$T_{p}(A|B)= \frac{A^{\frac{1}{2}}(A^{\frac{-1}{2}}BA^{\frac{-1}{2}})^{p}A^{\frac{1}{2}}-A}{p}$for $p\in(0, 1]$

and $T_{p}(A|B)$ can be written by using the notion of $A\# pB$ as follows:(1.1’)$T_{p}(A|B)= \frac{A\# pB-A}{p}$The relative operator entropy(1.2)$S^{\Lambda}(A|B)$

for $p\in(0,1]$ .

in [3] is defined by$\hat{S}(A|B)=A^{\frac{1}{2}}(\log A^{\frac{-- 1}{2}}BA^{\frac{-1}{2}})A^{\frac{1}{2}}$as an extension of [15].
On the other hand, the genaralized Kantrovich constant $K(p)$ is defined by(1.3)

for any real number$K(p)= \frac{(h^{p}-h)}{(p-1)(h-1)}(\frac{(p-1)}{p}\frac{(h^{p}-1)}{(h^{p}-h)})^{p}$$p$

and $h>1$ . Also $S(p)$ is defined by(1.4)$S(p)= \frac{h^{A}\overline{h}\mathrm{P}\overline{-1}}{e\log h\overline{h}^{L}\mathrm{p}-\overline{1}}$for any real number . In particular$p$$S(1)>1$ is well known$S(1)= \frac{h^{\frac{1}{h-1}}}{e\log h^{\frac{1}{h-1}}}$is said to be the Specht ratio and90
REVERSE INEQUALITIES ASSOCIATED WITH TSALLIS RELATIVE OPERATOR ENTROPY

Theorem A. Let be strictly positive operator satisfying $MI\geq A\geq mI>0_{f}$ where$M>m>0$ . Put $h= \frac{M}{m}>1$ . Then the following inequalities hold:$A$(1.5)

for any $1\geq p>0$ .

$(Ax, x)^{p}\geq(A^{p}x, x)\geq K(p)(Ax, x)^{p}$(1.6)$S(1)\Delta_{x}(A)\geq(Ax, x)\geq\triangle_{x}(A)$(1.7)$K(p)\in(0, 1]$(1.8)$K(0)=K(1)$ $=1$ .

(1.9)$S(1)=e^{K’(1)}=e^{-K’(0)}$.

for $p\in[0, 1]$ .
.

where the deter minant
for strictly positive operator
by $\Delta_{x}(A)=\exp\langle((\log A)x, x)\rangle$ and (1.6) is shown in [4],$\triangle_{x}(A)$$A$

at a unit vector$x$

is defined(1.8) and (1.9) of Theorem A are shown in [8, Proposition 1] and (1.7) is shown in [9].

52 Two reverse inequalities involving Tsallis relative operator entropy$T_{p}(A|B)$

via generalized Kantorovich constant $K(p)$At first we shall state the following two reverse inequalities involving Tsallis relativeoperator entropy $T_{p}(A|B)$ via generalized Kantorovich constant $K(p)$ .
Theorem 2.1. Let and be strictly positive operators such that Mil $\geq A\geq m_{1}I>0$
and $p\in(0, 1]$ . Let,and $M_{2}I\geq B\geq m_{2}I>0$ . Put(I be norm alized positive linear map on $B(H)$ . Then the following inequalities hold:
$A$

$B$

$m= \frac{m_{2}}{M_{1}}f$(2.1)$( \frac{1-K(p)}{p})$$M= \frac{M_{2}}{m_{1}}$$h= \frac{M}{m}=\frac{M_{1}M_{2}}{\overline{m_{1}}m_{2}}>1$

I $(A)\beta_{p}\Phi(B)+\Phi(T_{p}(A|B))\geq T_{p}(\Phi(A)|\Phi(B))\geq\Phi(T_{p}(A|B))$

and(2.2)$F(p)\Phi(A)+\Phi(T_{p}(A|B))\geq T_{p}(\Phi(A)|\Phi(B))\geq\Phi(T_{p}(A|B))$

where $K(p)$ is the generalized Kantorovich constant defined in (1.3) and$F(p)= \frac{m^{p}}{p}(\frac{h^{p}-h}{h-1})(1-K(p)^{\frac{1}{p-1}})\geq 0$.

Remark 2.1. We remark that the second inequality of (2.1) of Theorem 2.1 is shown in[6] along [3] and the first one of (2.1) is a reverse one of the second one and also the second
inequality of (2.2) is as the same as the second one in (2.1) and the first one of (2.2) is areverse one of the second one. We shall give simple proofs of (2.1) and (2.2) including its
reverse inequality, respectively, via generalized Kantorovich constant $K(p)$ in (1.3)

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REVERSE INEQUALITIES ASSOCIATED WITH TSALLIS RELATIVE OPERATOR ENTROPY

We state the following result to prove Theorem 2.1.
Proposition 2.2. Let h $>1$ and let $g(p)$ be defined by:

for $p\in[0, 1]$ .

$g(p)= \frac{h^{p}-h}{h-1}+(1-p)(\frac{h^{p}-1}{p(h-1)})^{\overline{\mathrm{p}}-\overline{1}}B$

Then the following results hold:(i)$g(0)= \lim_{parrow 0}g(p)=0$.

(ii)$g(p)= \frac{h^{p}-h}{h-1}(1-K(p)^{\frac{1}{\mathrm{p}-1}})\geq 0$(iii)$g’(0)= \lim_{parrow 0}g’(p)=\log S(1)$$\lim=\log S(1)\underline{g(p)}$(iv)

for all $p\in[0, 1]$ .

.

.

$parrow 0p$

Also we need the following result to prove Theorem 2.1.
be strictly positive operators on a Hilbert space $H$ suchand,that $M_{1}I\geq A\geq m_{1}I>0$ cvnd $M_{2}I\geq B\geq m_{2}I>0$ . Put. Let $p\in$ $(0, 1)$ and also let (I be normalized positive linear map onTheorem B. Let$A$and$B$$m= \frac{m2}{M_{1}}$$M= \frac{M_{2}}{m_{1}}$$h= \frac{M}{m}=\frac{M_{1}M_{2}}{m_{1}m_{2}}>1$$B(H)$ .

Then the following inequalities hold:(i)$\Phi(A)\#_{\mathrm{P}}\Phi(B)\geq\Phi(A\# pB)$(ii)$\Phi(A)\# p\Phi(B)\geq\Phi(A\#_{\mathrm{P}}B)\geq\Phi(A)\# p\Phi(B)-f(p)\Phi(A)$where$\geq K(p)\Phi(A)\#_{\mathrm{P}}\Phi(B)$$f(p)=m^{p}|| \frac{h^{p}-h}{h-1}+(1-p)(\frac{h^{p}-1}{p(h-1)})^{\overline{\rho}-\overline{1}}|\mathrm{A}|$and $K(p)$ is defined in (1.3).

The right hand side inequalities of (i) and (ii) of Theorem
and the left hand side of (i) is well known [13].

$\mathrm{B}$follow by [14, Corollary 3.5]

Q3 Two results by Furuichi-Yanagi-Kuriyama which are useful to prove ourresults in \S 6matrix. A matrix $X$ is said to be strictly
Throughout this section, we deal withpositive definite matrix (denoted by $X>0$ ) if $X$ is positide definite and invertible. Let
and be positive definite matrices. Tsallis relative entropy $D_{p}(A||B)$ in Furuichi-Yanagi$n\mathrm{x}$$n$

$A$

$B$

Kuriyama [5] is defined by(3.1)$D_{p}(A||B)= \frac{\mathrm{T}\mathrm{r}[A]-\mathrm{T}\mathrm{r}[A^{\mathit{1}-p}B^{p}]}{p}$Umegaki relative entropy $S(A, B)$ in [16] is defined by

for $p\in(0,1]$ .

92
REVERSE INEQUALITIES ASSOCIATED WITH TSALLIS RELATIVE OPERATOR ENTROPY[(3.2)$S(A, B)=\mathfrak{R}$ $A(\log$forA-log $B)$ ]$A$, $B>0$ .

Theorem C. (Genaralized Peierls-Bogoliubov inequality [5]) Let$p\in(0, 1]$ . Then the following inequality holds:(3.3)$D_{p}(A||B) \geq\frac{\mathrm{T}\mathrm{r}[A]-(\mathrm{T}\mathrm{r}[A])^{1-\mathrm{p}}(\mathrm{R}[B])^{\mathrm{p}}}{p}$$A$, $B>0$ and also let.

Theorem D[5]. Let A, B $>0$ . The following inequality holds:

-Tr

(3.4)

for $p\in(0,1]$ .

$[\mathrm{T}_{\mathrm{p}}(A|B)]\geq D_{p}(A||B)$

We remark that (3.4) implies
and [5].

$-\mathrm{T}\mathrm{r}[\hat{\mathrm{S}}(A|B)]$$\geq S(A, B)$

which is well known in [11],[12],[2]Q4 A result which unifies Theorem C and Theorem D in \S 3

Also throughout this section, we deal with $n><n$ matrix. In this section, we shall state
the following Proposition which unifies Theorem and Theorem in \S 3.
$\mathrm{D}$$\mathrm{C}$$\mathrm{E}$Proposition E. Let A, B $>0$ and also let p $\in(0,$1]. Then the following inequalities
hold:
Tr[(l $-p)A+pB$ ](4.1)$\geq$$($Tr$[A])^{1-p}(\mathrm{T}\mathrm{r}[B])^{p}$$\geq \mathrm{T}\mathrm{r}[A^{1-p}B^{p}]$$\geq \mathrm{T}\mathrm{r}[A\# pB]$.

Proposition F. Let A, B $>0$ and also let p $\in(0,$ 1]. Then the following inequalities
hold:
-Tr

(4.5)

$[\mathrm{T}_{\mathrm{p}}(A|B)]$$\geq D_{\mathrm{p}}(A||B)$$\geq\frac{\mathrm{T}\mathrm{r}[\mathrm{A}]-(\mathrm{T}\mathrm{r}[A])^{1-\mathrm{p}}(^{r}\mathrm{b}[B])^{\mathrm{p}}}{p}$$\geq \mathrm{T}\mathrm{r}[A -B]$.

Needless to say, the first inequality of (4.5) of Proposition is just (3.4) of Theorem
and the second one of (4.5) is just (3.3) of Theorem , and also Proposition is nothing
but another expression form of Proposition F.
$\mathrm{D}$$\mathrm{F}$$\mathrm{E}$$\mathrm{C}$

Proposition$\mathrm{F}$yields the following result by putting$parrow \mathrm{O}$.

Proposition G. Let A, B $>0$ . Then the following inequalities hold:
(4.6)

-Tr

$[\hat{\mathrm{S}}(A|B)]\geq S(A, B)$93
REVERSE INEQUALITIES ASSOCIATED WITH TSALLIS RELATIVE OPERATOR ENTROPY$\geq$$\geq$

Tr [ ($A$$\log$

Tr[A] $-\log$ Tr[B])]

Tr $[A-B]$ .

Q5 Related counterxamples to several questions caused by the results in \S 4matrix too. We shall give related
Also throughout this section, we deal withcounterxamples to several questions caused by the results in Q4$n\mathrm{x}$$n$

Remark 5.1. The following matrix inequality (AG) is quite well known as the matrix
version of (4.2) and there are a lot of references (for example, [13],[7]):$(1-p)A+pB\geq A\#_{P}B$ holds for $A$ , $B>0$ and $p\in(0, 1]$ .

(AG)

Suggested by the matrix inequality (AG), the second inequality and the third one oninequality $(4,1)$ of Proposition , we might be apt to suppose that the following matrix
inequalities as more exact precise estimation than (AG) : let , $B>0$ and $p\in(0, 1]$ ,$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$$\mathrm{E}$$A$$(1-p)A+pB\geq B^{E}2A^{1-p}B^{E}2\geq A\# pB$$(\mathrm{A}\mathrm{G}- 1^{7})$and$(1-p)A+pB\geq A^{\underline{1}-A^{1-}}2B^{p}A^{--_{2}\mathrm{A}}\geq A\# pB$$(\mathrm{A}\mathrm{G}- 2^{7})$.

But we have the following common counterexam ple to (AG).) andRemark 5.2. (i).

If$A$

and

$B$$(\mathrm{A}\mathrm{G}- 2^{7})$.

are positive definite matrices and $p\in(0, 1]$ , then thefollowing inequality $holds$ :(5.2)$D_{p}(A||B)\geq ib[A -B]$.

We remark that (5.2) is shown in the proof of [5, (1) of Proposition 2.4] and the second
inequality and the third one of (4.5) of Proposition yield the inequality (5.2), that is, thesecond inequality and the third one of (4.5) of Proposition are somewhat more precise$\mathrm{F}$$\mathrm{F}$estimation than (5.2).
(ii). Also we recall the following result [1, Problem IX.8.12]:

If

$A$

and

$B$

are strictly positive matrices, then the following inequality holds:(5.3)$\mathrm{T}\mathrm{r}[A(\log A-\log B)]\geq \mathrm{T}\mathrm{r}[A-B]$.

We remark that the second inequality and the third one of (4.6) of Proposition imply[$A(\log$ A-log $B)$ ], that is, the second inequality and the third one(5.3) sinceof (4.6) are somewhat more precise estimation than (5.2).
$\mathrm{G}$$S(A, B)=\mathrm{T}\mathrm{r}$94
REVERSE INEQUALITIES ASSOCIATED WITH TSALLIS RELATIVE OPERATOR ENTROPY

Suggested by (5.3), we might be apt to expect that the following matrix inequality:(5.3-1?)$A^{\frac{1}{2}}(\log A-\log B)A^{\frac{1}{2}}\geq A-B$.

so that it turns out that (5.3-1?) does not hold.
reverse inequalities associated withQ6. Twovia generalized Kantorovich constant $K(p)$$-\mathrm{T}\mathrm{r}[T_{p}(A|B)]$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$and $D_{p}(A||B)$reverse inequalAs an application of Theorem 2.1 we shall show the following twoities associated with $-Tr[T_{p}(A|B)]$ and $D_{p}(A||B)$ via generalized Kantorovich constant$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$$K(p)$ .

Theorem 6.1.
Letand $B$ be strictly positive definite matrices such that $M_{1}I\geq A\geq m_{1}I>0$ and. Then
andand,$M_{2}I\geq B\geq m_{2}I>0$ . Put$A$$m= \frac{m_{2}}{M_{1}}$$M= \frac{M_{2}}{m_{1}}$$h= \frac{M}{m}=\frac{M_{1}M_{2}}{m_{1}m_{2}}>1$$p\in(\mathrm{O}, 1]$

the following inequalities hold:(6.1)$( \frac{1-K(p)}{p})(’\mathrm{b}[A])^{1-\mathrm{p}}(\mathrm{T}\mathrm{r}[B])^{\mathrm{p}}+D_{p}(A||B)$$\geq-\mathrm{T}\mathrm{r}[T_{p}(A|B)]$$\geq D_{p}(A||B)$(6.2)$F(p)(’\mathrm{b}[A])+D_{p}(A||B)$$\geq-\mathrm{T}\mathrm{r}[\mathrm{T}_{p}(A|B)]$$\geq D_{p}(A||B)$

where $K(p)$ is the generalized Kantorovich constant defined in (1.3) and.

$F(p)= \frac{m^{p}}{p}(\frac{h^{p}-f\iota}{h-1})(1-K(p)^{\frac{1}{\mathrm{p}-1}})\geq 0$

Corolleiry 6.2. [10] Let and $B$ be strictly positive definite matrices such that $M_{1}I\geq$. Then the following inequality$A\geq m_{1}I>0$ and $M_{2}I\geq B\geq m_{2}I>0$ . Put
hold:
$A$

$h= \frac{M_{1}M_{2}}{m_{1}m_{2}}>1$(6.5)$\log S(1)\ulcorner \mathrm{R}[A]+S(A, B)$$\geq-\mathrm{b}[\hat{S}(A|B)]$$\geq S(A, B)$

where $S(1)$ is the Specht ratio defined in (1.4) and the first inequality is the reverse one of
the second in equalty.

$\theta 5$

REVERSE INEQUALITIES ASSOCIATED WITH TSALLIS RELATIVE OPERATOR ENTROPY

The complete paper with proofs will appear elsewhere.
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Department of Mathematical Information Science
Tokyo University of Science
1-3 Kagurazaka, Shinjukuku

Tokyo 162-8601
Japan-mail: furuta(Ors.kagu.
$\mathrm{e}$$\mathrm{t}\mathrm{u}\mathrm{s}$.ac.jp