数理解析研究所講究録
第 1632 巻 2009 年 61-67

61

Further extension of an order preserving operator inequality and its application

Takayuki Furuta
Department of Mathematical Information Science
Tokyo University of Science

An operator is said to be positive (denoted by $T\geq 0$ ) if $(Tx, x)\geq 0$ for all $x\in H$ , andis said to be strictly positive (denoted by $T>0$ ) if $T$ is positive and invertible.
The celebrated L\"owner-Heinz inequality as serts that if $A\geq B\geq 0$ , thenholds
for any $\alpha\in[0,1]$ , but $A^{p}\geq B^{p}$ does not always hold for $p>1$ . From this point of view,
we shall show an order preserving operator inequality in Theorem 2.3 and we shall state animplication among Theorem 2.3 and the previous known results.
$T$

$T$

$A^{\alpha}\geq B^{\alpha}$Theorem 2.3. Let $A\geq B\geq 0$ with $A>0,$ $t\in[0,1]$ and. Then the following inequality holds for $r\geq t$ :$p_{1},p_{2},$ $\ldots.,p_{2n}\geq 1$for natural number$n$$A^{1-t+r}\geq$$\{$$A^{r} \tau[.\cdots\cdots\cdot\cdots\underline{A}B^{p_{1}}\frac{\llcorner A^{\overline{\tau}^{tt}}\{A^{p}}{arrow A\overline{\tau}_{ntime\S a}^{t}-}..\frac{\cdots\cdots\cdots\cdot[A^{\overline{\tau}^{t}}\{A\#(A^{\overline{\tau}^{\underline{t}}}}{ndA\#_{n-1tim6gby}}\frac{A^{\overline{\tau}’})^{p_{2}}A^{z}\}^{ps}A^{\overline{\tau}^{t}}]^{p_{4}}A^{w}\ldots\ldots\ldots\ldots.A^{\overline{\tau}^{t}}\iota.\iota}{arrow A^{=}r_{ntim\epsilon sandA}^{t}f_{n-1t1mesbyturns}}turns]^{p_{n_{A}}^{-}}}^{\text{楠^{}-l+}h}|.|+r$where$q[2n]=\{c$..... $[\{[(p_{1}-t)p_{2}+t]p_{3}-t\}p_{4}+t]p_{5}-\ldots\ldots-t\}p_{2n}+t$ .

$-\overline{- t}$$\lrcorner$

and $talternatelyntim\infty$ appear

Remark 1. Consider the following operator function. $f($ ロ defined by$)$$f($ ロ $)=\{$ $A^{\xi}[$$\underline{\underline{cA^{=}\Gamma^{t}\{A^{t}F\ldots\ldots\ldots\ldots\ldots}\ldots.[A^{\overline{\tau}^{t}}\{A}-t\underline{\not\in(A^{\overline{\dot{\tau}}’}\lrcorner}$$arrow ATn$ times and $A\# n-1$ times by turns

ロ$\underline{A\overline{\dot{T}}^{tt})^{p_{2}}AF\}^{p_{3}}A^{\overline{\tau}^{l}}]^{P4}A^{f}\ldots\ldots\ldots\ldots.A^{\overline{\dot{\tau}}^{t}}}]^{p_{2n}}A}^{\frac{1-t+r}{q[2n]+r-i}}$$arrow A^{\vee}\tau^{t}n$times and $Afn-1$ times by tums

Theorem 2.3 asserts that $f(A^{p_{1}})\geq f(B^{p_{1}})$ holds for any
whenever $A\geq B\geq 0$ with
$A>0$ under some suitable $conditions_{f}$ that is, (ロ) can be considered as order preserving
operator function. Therorem 2.3 is further extension of the following previous one: if$A\geq B\geq 0$ with $A>0$ , thenfor $t\in[0,1]$ and $p\geq 1,$
holds for $r\geq t$ and $s\geq 1$ , in particular,
for $p\geq 1$ and $r\geq 0$ . Finallyan application of Theorem 2.3 is shown.
$p_{1}\geq 1$$f$$A^{1-t+r} \geq\{A^{\frac{r}{2}}(A^{\frac{-l}{2}}B^{p}A^{\frac{-\iota}{2}})^{s}A^{\frac{r}{2}}\}\frac{1-t+r}{(p-t)\epsilon+r}$$A^{1+r}\geq(A^{\frac{r}{2}}B^{p}A^{\frac{r}{2}})^{\frac{1}{p}\pm}+^{\frac{r}{r}}$.

62\S 1 Introduction$T$

An operator $T$ is said to be positive (denoted by $T\geq 0$ ) if $(Tx, x)\geq 0$ for all $x\in H$ , andis said to be stmctly positive (denoted by $T>0$ ) if is positive and invertible.
$T$Theorem LH (L\"owner-Heinz inequality, denoted by (LH) briefly).

If $A\geq B\geq 0$ holds, then$A^{\alpha}\geq B^{\alpha}$for any$\alpha\in[0,1]$.

(LH)

Although (LH) asserts that $A\geq B\geq 0$ ensures
for any $\alpha\in[0,1]$ , unfortunately
does not always hold for $p>1$ . The following result has been obtained from thispoint of view.
$A^{\alpha}\geq B^{\alpha}$$A^{p}\geq B^{p}$

Theorem F.

If $A\geq B\geq 0$ , then for each(i)$r\geq 0$,$(B^{L}2A^{p}B^{\frac{r}{2}})^{\frac{1}{q}}\geq(B^{\frac{r}{2}}B^{p}B^{\frac{r}{2}})^{\frac{1}{g}}$and(ii)$(A^{\frac{r}{2}}A^{p}A^{\frac{r}{2}})^{\frac{1}{q}}\geq(A^{\frac{r}{2}}B^{p}A^{r}2)^{\frac{1}{q}}$hold for $p\geq 0$ and $q\geq 1$ with $(1+r)q\geq p+r$ .

The original proof of Theorem is shown in [Fl], an elementary one-page proof is in[F2] and alternative ones are in [MF],[KI]. It is shown in [Tl] that the conditions
andin FIGURE 1 are best possible.
$F$

$p,$

$q$

$r$

The original proof of Theorem
is in [F3], and an alternative one is in [MF-K]. Anelementary one-page proof of (1.1) is in [F4]. We mention that further extensions of Theorem and related results to Theorem are in [MF-K-N],[F5],[F-H-I],[F-Y-Y], [K2] andetc. It is shown in [T2] that the exponent valueof the right hand of (1.1)
$G$

$G$

$F$

$\frac{1-t+r}{(p-t)s+r}$

is best possible and alternative ones are in [MF-M-N],[Y]. It is known that the operator
inequality (1.1) interpolates Theorem and an inequality (see Theorem AH under below at$F$

634 page) equivalent to the main result of Ando-Hiai$t\in[0,1]$ .

\S 2 FUrther extension of Theorem$\log$

majorization [A-H] by the parameter$G$

Corollary 2.4 [F6]. If $A\geq B\geq 0$ with $A>0,$ $t\in[0,1]$ and$p_{1},p_{2},p_{3},p_{4}\geq 1$,$A^{1-t+r}\geq\{A^{\frac{r}{2}}[A^{\frac{-t}{2}}\{A^{\frac{t}{2}}(A^{\frac{-t}{2}}B^{p_{1}}A^{\frac{-t}{2}})^{p_{2}}A^{\frac{t}{2}}\}^{p_{3}}A^{\frac{-t}{2}}]^{P4}A^{\frac{r}{2}}\}^{\frac{1-\ell+r}{\lceil\{(p_{1}-t)p_{2}+t\}p_{3}-t|p_{4}+r}}$

holds for $r\geq t$ .

64

Corollary 2.4 obtained by Theorem 2.3 and previous known results

Corollary 2.4. If $A\geq B\geq 0$ with $A>0,$ $t\in[0,1]$ and$p_{1},p_{2},p_{3},p_{4}\geq 1$,$A^{1-t+r}\geq\{A^{\frac{r}{2}}[A^{\frac{-t}{2}}\{A^{\frac{t}{2}}(A^{\frac{-t}{2}}B^{p_{1}}A^{\frac{-t}{2}})^{p_{2}}A^{\frac{t}{2}}\}^{p_{3}}A^{\frac{-t}{2}}]^{p_{4}}A^{\frac{r}{2}}\}^{\frac{1- t+r}{|\{(p_{1}-t)p_{2}+t\}p_{3}-t|p_{4+}r}}$holds for $r\geq t$ .
$p_{2}=p_{3}=1$$\downarrow$

Theorem G. If $A\geq B\geq 0$ with $A>0$ , then for $t\in[0,1]$ and $p\geq 1$ ,$A^{1-t+r} \geq\{A^{\frac{r}{2}}(A^{\frac{-t}{2}}B^{p}A^{\frac{-t}{2}})^{s}A^{\frac{r}{2}}\}\frac{1-t+r}{(p-t)s+r}$

holds for $r\geq t$ and $s\geq 1$ .
$t=0$ and $s=1$$t=1$ and $r=s$$\swarrow$$\searrow$

Theorem AH. If $A\geq B\geq 0$ with $A>0\Rightarrow$

Theorem F. $A\geq B\geq 0\Rightarrow$$A^{r}\geq\{A^{L}2(A^{\frac{-1}{2}B^{p}A^{\frac{-1}{2})^{r}A^{\frac{r}{2}}\}^{\frac{1}{p}}}}$$A^{1+r}\geq(A^{\frac{r}{2}}B^{p}A^{\frac{r}{2}})^{\frac{1+r}{p+r}}$

for$p\geq 1$

for $r,p\geq 1$ .

and $r\geq 0$ .

$\Uparrow$

Theorem AH-L. For every $A,$ $B\geq 0$ and $0\leq\alpha\leq 1$ ,$\Uparrow$$(A\neq\alpha B)^{r}\succ A^{r}\#\alpha B^{r}(\log)$

Theorem F.

If $A\geq B\geq 0$ , then for each(i)$r\geq 0$,$(B^{\frac{r}{2}}A^{p}B^{\frac{r}{2}})^{\frac{1}{g}}\geq(B^{\frac{r}{2}}B^{p}B^{\frac{r}{2}})^{\frac{1}{q}}$and(ii)$(A^{\frac{r}{2}}A^{p}A^{\frac{r}{2}})^{\frac{1}{q}}\geq(A^{\frac{f}{2}}B^{p}A^{\frac{r}{2}})^{\frac{1}{q}}$hold for $p\geq 0$ and $q\geq 1$ with $(1+r)q\geq p+r$ .

for$r\geq 1$.

65\S 3 IFUrther extension of an operator function implying Theorem$G$

Theorem H. If $A\geq B\geq 0$ with $A>0$ , then for $t\in[0,1]$ and $p\geq 1$ ,$F_{A,B(r,s)=A^{\frac{-r}{2}\{A^{\frac{r}{2}}(A^{\frac{-t}{2}}B^{p}A^{\frac{-t}{2}})^{s}A^{\frac{r}{2}}\}}} \frac{1-t+r}{(p-t)s+r}A^{\frac{-r}{2}}$is a decreasing function for $r\geq t$ and $s\geq 1$ , and $F_{A,A}(r, s)\geq F_{A,B}(r, s)$ holds, that is,(1.1)$A^{1-t+r}\geq\{A^{\frac{r}{2}}(A^{\frac{-t}{2}}B^{p}A^{\frac{-\ell}{2}})^{s}A^{\frac{f}{2}}\}^{\frac{1-t+r}{(p-t)s+r}}$

holds for $t\in[0,1],$ $p\geq 1,$ $r\geq t$ and $s\geq 1$ .
in Theorem is an operator function implying (1.1) in Theorem G. The original proof of Theorem is in [F3], and an alternative one is in [MF-K]. Further extensionsof Theorem and related results are in [MF-K-N],[F5],[F-H-I],[F-Y-Y], [K2] and etc.
$F_{A,B}(r, s)$

$H$

$H$

$H$

We shall state further extension of Theorem$H$

as an application of Theorem 2.3.

Theorem 3.1 [F8]. Let $A\geq B\geq 0$ with $A>0,$ $t\in[0,1]$ andnumber$n$$p_{1},p_{2},$$\ldots.,p_{2n}\geq 1$. Thenfor $natum|$(3.1)$G_{A,B}[r,p_{2n}]$$=A\overline{\tau}^{\underline{r}}\{$$A9[ arrow turn8arrow turns\frac{A^{\sim}\overline{\tau}^{t}\{A\xi\ldots\ldots\ldots\ldots\ldots..[A^{-r}\{A\tau(A\tau^{\wedge}- lt-t}{A\overline{\tau}_{ntimegandA}^{t}i_{n-1timesb_{Y}}}\frac{A\overline{Y}^{\underline{t}})^{p}2A\not\in\}^{pa}A\overline{r}^{t}]^{p_{4}}A^{t}\tau\ldots\ldots\ldots..A\overline{\tau}^{f}}{A\overline{7}^{\underline{t}}ntime\text{\’{e}} andAf_{n-1time\S by}}B^{p}1.]^{p}A\tau\}^{\frac{1+-f}{\eta[2n]+r-t}}A\overline{7}^{-}$

is a decreasing function of $p_{2n}\geq 1$ and $r\geq t$ , and the following inequality holds$G_{A,A}[r,p_{2n}]\geq G_{A,B}[r,p_{2n}]$,that is,$A^{1-t+r}\geq$$\{$ $A^{}$$\underline{A^{-\tau})^{p_{2}}A^{f}\}^{p_{3}}A7]^{p_{4}}A^{f}-t-\underline{t}\ldots\ldots\ldots..\}A^{\overline{\tau}^{\iota}}}]^{p_{2n}}$$[arrow A\overline{\tau}_{ntimesandA}^{l}\not\in\underline{\underline{A^{\overline{\tau}^{l}}\{A^{t}f\ldots\ldots\ldots\ldots}\ldots\underline{..[A\overline{\tau}^{t}\{A^{\xi}(A\overline{\tau}^{f_{\lrcorner}}}}n-1timesbyturnsB^{p_{1}}$$arrow A^{=}\tau’n$

where $q[2n]$ is defined by (2.2).

times and A\S $n-1$ times by tums$z\}^{\frac{1-l+}{q[2n|+r-\cdot t}}$$A^{r}$$(3.2)$

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Corollary 3.2 [F8]. If $A\geq B\geq 0$ with $A>0,$ $t\in[0,1]$ and $p_{1},p_{2},p_{3},p_{4}\geq 1$ ,$G_{A,B}[r,p_{4}]$$=A^{\frac{-r}{2}}\{A^{\frac{r}{2}}[A^{\frac{-t}{2}}\{A^{\frac{\ell}{2}}(A^{\frac{-t}{2}B^{p_{1}}A^{\frac{-l}{2}}})^{p_{2}}A^{\frac{\ell}{2}}\}^{p_{3}}A^{\frac{-t}{2}}]^{\rho_{4}}A^{\frac{r}{2}}\}^{\frac{1-\ell+r}{|\{(p_{1}-t)p_{2}+t\}p_{3}-t|p_{4+\Gamma}}}A^{\frac{-r}{2}}$is a decreasing function of $p_{4}\geq 1$ and $r\geq t$ , and the following inequality holds$G_{A,A}[r,p_{4}]\geq G_{A,B}[r,p_{4}]$, that is,$A^{1-t+r}\geq\{A^{\frac{r}{2}}[A^{\frac{-t}{2}}\{A^{\frac{t}{2}}(A^{\frac{-\ell}{2}}B^{p_{1}}A^{\frac{-t}{2}})^{p_{2}}A^{\frac{f}{2}}\}^{p_{3}}A^{\frac{-t}{2}}]^{p_{4}}A^{\frac{r}{2}}\}^{\frac{1-t+r}{[\{(p_{1}-t)p_{2}+t\}p_{3}-\ell]p_{4}+\tau}}$holds for $t\in[0,1],$ $r\geq t$ and$p_{1},p_{2},p_{3},p_{4}\geq 1$.

Remark 3.1. Theorem 3.1 yields Corollary 3.2 by putting $n=2$ and also Corollary
3.2 yields Theorem by putting $p_{2}=p_{3}=1$ .
$H$\S 4 Satellite inequalities as an application of Theorem 3.1
Theorem 4.1 [F8]. If $A\geq B\geq 0$ with $A>0,$ $t\in[0,1]$ andfollowing inequality holds:$p_{1},p_{2},$$\ldots.,p_{2n}\geq 1$, then the$A\geq B$$\geq\{A^{\iota\frac{1}{(p_{1}-t)p_{2}+t}}2(A^{\frac{-t}{2}}B^{p_{1}}A^{\frac{- t}{2}})^{p_{2}}A^{\frac{t}{2}}\}$$\geq\{A^{\frac{t}{2}}[A^{\frac{-t}{2}}\{A^{\frac{t}{2}}(A^{\frac{-\ell}{2}}B^{p_{1}}A^{\frac{-t}{2}})^{p_{2}}A^{\frac{t}{2}}\}^{p_{3}}A^{\frac{-\ell}{2}}]^{p_{4}}A\S\}^{\frac{1}{\{|(p_{1}-t)p_{2}+t|p_{3}-t\}p_{4}+t}}$$\geq$$\geq[_{\underline{\underline{c}}}A^{1}2\{A^{\frac{-t}{2}}[A^{\frac{t}{2}}\ldots\ldots\ldots..[A^{\frac{-t}{2}}\{A^{\frac{t}{2}}(A^{\frac{-t}{2}}B^{p_{1}}A^{\frac{-t}{2}})^{p_{2}}A^{\frac{t}{2}}\}^{p_{3}}A^{\frac{-t}{2}}]^{p_{4}}\ldots\underline{.A^{\frac{t}{2}}]^{p_{2n-1}}A^{\frac{-t}{2}}\}_{\lrcorner}^{p_{2n}}A^{\frac{t}{2}}}]-\iota-t\underline{\lrcorner}\underline{\underline{\llcorner}}T^{1}q2\urcorner n$$arrow A^{-}T$

and$A\xi$

alternately$n$times$arrow A\tau$and$A^{t}f$altemately$n$times

where $q[2n]$ is defined in (2.2).

Remark 4.1. Corollary 2 in \S 3.2.5 of [F5] states that if $A\geq B>0$ , then$A \geq B\geq\{A^{\frac{t}{2}}(A^{\frac{-t}{2}}B^{p}A^{\frac{-t}{2}})^{s}A^{\frac{t}{2}}\}\frac{1}{(p-\ell)*+t}$holds for each $t\in[0,1]$ andand Theorem 4.1 is further extension of (4.2).

$p,$$s\geq 1$(4.2)$(4.1)$

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Department of Mathematical Information ScienceFaculty of Science
Tokyo University of Science
1-3 Kagurazaka, Shinjukuku
Tokyo 162-8601
Japan

email:furuta@rs.kagu.tus.ac.jp