数理解析研究所講究録1080 巻 1999 年 156-159

156

AN APPLICATION OF ALUTHGE TRANSFORM TO
PUTNAM INEQUALITY FOR LOG-HYPONORMAL OPERATORS

藤井正俊$\mathrm{M}\wedge \mathrm{S}\Lambda \mathrm{T}\mathrm{o}\mathrm{s}\mathrm{I}\mathrm{I}\mathrm{I}\Gamma^{\mathrm{I}}\mathrm{U}\mathrm{J}[\mathrm{I}$. In this note, we give a shorl proof t,o the Putnam inequality for log-hyponormal
and $\log$ -hyponormal, i.e., $\log T^{\cdot}T\geq\log TT$ ,operators due to Tanahashi: If 7’ ishenA$\mathrm{D}\mathrm{S}\uparrow \mathrm{R}\mathrm{A}\mathrm{C}\mathrm{T}$$\mathrm{i}\mathrm{n}\mathrm{v}\mathrm{o}\mathrm{l}\mathrm{t}.\mathrm{i}\mathrm{b}[\mathrm{e}$$\mathrm{t}$$|| \log?^{\mathrm{s}_{7’-}}’\log T\tau^{\mathrm{s}}||\leq\frac{1}{\pi}\int\int_{\sigma(T)}T^{-\mathrm{l}}drd\theta$

or,

$T$ .
I , is based on his original idea that. the log-hyponormality
is he spectrum
whereis regarded as -liyponorrnalil, .
$\sigma(T)$$\mathrm{t}$$0$$\mathrm{t}$$\mathrm{y}$Furuta1. Introduction. Afterwas originated by Furuta [12], see alsot,ed{,he[5,13,16,18], weto study it underchaotic order $A>>B$ , i.e., $\log A\geq\log B$ , for. We finally cllaracterized $A>>B$ by a Furuta-type
posit ive inverl,ible operat,ors: For $A,$ $D>0,$
inequalit,
only if$\mathrm{t},1\mathrm{l}\mathrm{e}$$\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{q}\mathrm{u}_{\mathrm{c}}\backslash 1\mathrm{i}\{,\mathrm{y}$$\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}_{\mathrm{e}}\gamma$$\Lambda_{\mathrm{c}}’\iota 11\mathrm{d}f\mathit{3}[11]$$\Lambda>>\mathcal{B}\mathrm{i}[’\iota\subset \mathrm{n}\mathrm{d}$$\mathrm{y}[6](\urcorner \mathrm{n}\mathrm{d}[7,81$(1)$(\Lambda^{\gamma}f\mathit{3}\mathrm{p}\Lambda^{\mathcal{T}})^{\frac{2r}{\rho\{2r}}\leq A^{2r}$.
for all
Furt hermore we[9.10]:$11\mathrm{o}\mathrm{l}\mathrm{e}1_{\mathrm{S}}$$p,$$\uparrow\cdot\geq 0$$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{t}\gamma\iota,\mathrm{e}\mathrm{d}$Tlleorem A.

$\Gamma’or$

a fixed $\delta>0,$

between$\dagger,1$$A^{\delta}\geq B^{\delta}$(2))$\mathrm{e}$Furuta inequality and Theorem A as follows

for$\Lambda,$$B\geq 0$

if and only for each$i,f$$r\geq 0$$\Lambda^{\mathrm{L}_{\frac{+2r}{\eta}}}\geq(\Lambda^{r}B^{\rho}A^{r})^{\frac{1}{q}}$holds for $p\geq 0$ and $q\geq 1$ with(.3)$(\delta-\vdash 2?\cdot)q\geq|^{y\dashv-2}r$$1l’\mathrm{e}$is.

$|_{11\mathrm{S}}\mathrm{t}$nole lhat (1) islhe case$\delta=1$.

$\mathrm{e}\mathrm{q}_{11\mathrm{i}\mathrm{V}_{\mathrm{C}}\gamma}1_{\mathrm{C}\mathrm{n}}\{,$The($\mathrm{t},0$$1_{\mathrm{o}\mathrm{m}_{\mathrm{c}}\urcorner}\mathrm{i}\cap$.

the case $\delta=0$ in Theorem A and the Furuta inequality
given by (3) is explained by the figure below:

157-hyponormality for invertible
From the viewpoint of this, Tanahashi [19] defined the$X^{*}X$$|X|$, and constructed the Putnam
is the squere root ofoperators by $|T|>>|T^{*}|$ , where-hyponormal operators:
inequality for$\log$$\log$Theorem T. Ifthen$T$

is an invertible $log$ -hyponormal operator, $i.e.,$ $\log T*T-\log\tau\tau*\geq 0$ ,,(4)$|| \log\tau^{*\tau}-\log T\tau^{*}||\leq\frac{1}{\pi}\int\int_{\sigma(T)}r-1drd\theta$

where$\sigma(T)$is the spectrum of $T$ .

It was conjectured from the Putnam inequality for p–hyponormal operators by Cho andItoh [3]:
If is a -hyponorlnal operator, i.e., $(T^{*}T)^{p}-(TT*)^{\mathrm{P}}\geq 0$ , then$T$$\mathrm{p}$(5)$||( \tau*\tau)^{p}-(T\tau*)^{p}||\leq\frac{p}{\pi}\int\int_{\sigma(}\tau)r^{2p-}1drd\theta$.

As a matter of fact, he understood (5) as follows:(6)$|| \frac{(T^{*}T)p-(T\tau^{*})^{p}}{p}||\leq\frac{1}{\pi}\int\int_{\sigma(\tau)}r^{2_{\mathrm{P}}1}-drd\theta$.

By taking $parrow\infty$ , he constructed Theorem and proved it by the idea developed in (5).
The purpose of this note is to continue his consideration directly. That is, we here propose
a straight and simple proof of Theorem which might be along with his-hyponormality as the -hyponormality. Our tool in this note is themight regards theAluthge transform which is grown up the p–hyponormality, see [1,2,4,14,15,20].
$\mathrm{T}$$\mathrm{T}$$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}_{0}\mathrm{n};\mathrm{T}\mathrm{a}\mathrm{n}\mathrm{a}\mathrm{h}\mathrm{a}\mathrm{S}\mathrm{h}\mathrm{i}$$0$$\log$

2. Preliminary.
For the sake of convenience, we cite the following characterization of chaotic order which
implies (1) by the help of the Furuta inequality [7], see also [8] and [9].

Theorem B. For $A,$ $B>0,$ $A>>B,$ $i.e.,$ $\log A\geq\log B$ , if and only if for any $\delta\in(0,1]$
there exists ansuch that$\alpha=\alpha_{\delta}>0$$(e^{\delta}A)^{\alpha}>B^{\alpha}$.

The essential part of Theorem is as follows: Ifthen there exists an $\alpha\in(0,1]$ such that$\mathrm{B}$$e^{\alpha A}>e^{\alpha B}$$(^{*})$$A$and $B$ are selfadjoint and $A>B$ ,.

It has the following simple proof: The assumption $A>B$ means that $A-B\geq\epsilon>0$ forsome . We here take $0<\alpha<\epsilon/(e^{||A||}+e^{||B||})$ and. Then we have$\epsilon$$\alpha\leq 1$$e^{\alpha A}-e^{\alpha B}= \alpha(A-B)+\sum_{n=2}\frac{\alpha^{n}}{n!}(A\infty)n-B^{n}$$\geq\alpha\epsilon+\alpha^{2}\sum_{=n2}^{\infty}\frac{\alpha^{n-2}}{n!}(A^{n}-B^{n})$$\geq\alpha\epsilon-\alpha|2|\sum_{n}\infty=2\frac{\alpha^{n-2}}{n!}(A^{n}-Bn)||$$\geq\alpha\epsilon-\alpha^{2}n=\sum^{\infty}\frac{1}{n!}(||A||^{n}+||B||^{n})2$$\geq\alpha(\epsilon-\alpha(e^{||A||}+e^{1})|B||)>0$.

158
Here we should note an interesting characterization of chaotic order recently obtainedby Yamazaki and Yanagida [22], which is associated with Kantorovich inequality and consequently Specht’s ratio, see [23].

3. Proof.

We begin with the Putnam inequality for -hyponormal operators; we turn it into thefollowing lemma via the L\"owner-Heinz inequality:
Lemma 1. If is a -hyponormal operator, then satisfies (6) for all $0<p\leq q$ . Consequently (4) holds for any $q- h\uparrow Jponomlal$ operators .
$\mathrm{p}$

$T$

$T$

$q$

$T$

for a positive invertible
The second half is ensur.ed.by the fact thatoperator .
Thus Lemma 1 suggests us to find a family $\{T_{q}; q>0\}$ of -hyponormal operators such-hyponormal operator $T$ . In this situation,
for a giventhat $||T_{q}-T||arrow 0$ as
the Aluthge transform completely responds to our demand. As a matter of fact, Tanahashi
prepared the following result in [20]:Lemma 2. If $T$ is an invertible $log$ -hyponormal operator with the polar decomposition $T=$$\tilde{T}=|T|^{q}U|T|^{1-q}$ is.
-hyponormal for$U|T|$ , then the Aluthge transform$\mathrm{u}-\lim_{tarrow 0}\frac{A^{t}-1}{t}=\log$

$A$

$A$

$q$

$\log$$qarrow \mathrm{O}$$0<q \leq\frac{1}{2}$$q$To prove Theorem , we take $T_{q}=|T|^{q}U|T|^{1-q}$ for $0<q< \frac{1}{2}$ . Since $\sigma(T_{q})=\sigma(T)$ for
all , it is compiete.
We finally give a short proof to Lemma 2 via Theorem APutting $p=2q$ and $r=1-q$ in (1), we have$\mathrm{T}$$q$$(T_{q}^{*}\tau_{q})1-q=(|T|^{1-q}U^{*}|T|2qU|T|^{1q}-)1-q$$=U^{*}(U|T|^{1-}qU^{*}|T|^{2q}U|\tau|^{1-q}U*)^{1}-qU$$=U^{*}(|\tau*|1-q|\tau|2q|\tau^{*}|^{1-}q)^{1q}-U$$\geq U^{*}|T^{*}|^{2(-}1q)U$$=|T|^{2(}1-q)$,

so that $(T_{q}^{*}Tq)2q\geq|T|^{2q}$ via the L\"owner-Heinz inequality by $1-q\geq q$ . On the other hand,
we have also$(T_{q}T_{q}^{*})^{q}=(|T|^{q}U|T|^{2}(1-q)U*|T|^{q})^{q}$$=(|T|^{q}|\tau*|^{2(}1-q)|\tau|^{q})^{q}$$\leq|T|^{2q}$.

Therefore it follows that$(\tau_{q}^{*}T_{q})^{q}\geq|T|^{2q}\geq(T_{q}T_{q}^{*})^{q}$,as desired.
Remark. (1) (5) is obtained by Putnam [17] for $p=1$ , Xia [21] for $\frac{1}{2}\leq p<1$ andCho-Itoh [3] for $0<p< \frac{1}{2}$ .
(2) The proof of Lemma 2 is available to show Tanahashi’s result [20; Theorem 4].
Acknowledgement. The authors would like to express their thanks to Prof. K.Tanahashi for his fascinating talk in the meeting of the Japan Mathematical Society at March
26, 1998. He also thanks to Prof. E.Kamei for his critical discussion.

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$A^{\delta}\geq B^{\delta}$

$p$

$p$

$F\alpha ru$$\mathrm{C}.\mathrm{R}$$7\dot{\mathrm{Y}}nt$$operat_{\mathit{0}}rsr$$\mathrm{J}.\mathrm{I}$

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