数理解析研究所講究録
第 1632 巻 2009 年 48-54

48

Generalized parallelogram law for operatorsand its application
左紅亮 (河南師範大学)

Hongliang Zuo (Henan Normal University)
藤井正俊(大阪教育大学)

Masatoshi Fujii (Osaka Kyoiku University)

Abstract.

scalars

$a$

.

$b$

The calssical Bohr inequality says that $|a+b|-\leq p|a|^{\wedge})+q|b|-$ for all

and

$p$

. $q>0$ with $1/p+1/q=1$ . In this note. we improve the accuracv

of the estimate given by the original Bohr inequalitv. Actuallv. we present:

If.4 and$B$

are operators on a Hilbert space and $t\neq 0$ , then$|A+B|^{\sim}+ \frac{1}{t}|t.4-B|^{-}=(1+t)|.4|^{-}+(1+\frac{1}{t})|B|^{2}$.

We discuss applications and further generalizations of it.

\S 1

Introduction

Let $H$ be a coniplex separable Hilbert spacv and $B(H)$ the algebra of all bounded operalors

on

$H$

. Denote by$|A|$

the absolute value operator of $A\in B(H):|.4|=(A‘ \mathcal{A})^{1/2}$ . where .4‘ istlie a ljoiiit operator of.4.

.

We say that .4 is a positive operator. if ( $.4_{A,X)}\geq 0$ for all $x\in H$ , denoted by $A\geq 0$ an$4\geq B$

if$A$$(\rfloor$

and $B$ are self-arljoint and.$4-B\geq 0$ .

The classical Bohr inequalitv for scalar asserts that for aiiy $n,$$b\in \mathbb{C}$and all positive

conjugate expoIients $p,$ $q\in R$ ,(0)$|a+b|^{2}\leq p|(\iota|^{2}+q|b|^{2}$svith equality if and only if $(1-p)a=b$ (See [1]).
In 2003, O. Hirzallah[4] proposed an operator version of Bohr inequality as follows:.

If $A,$ $B\in B(H)$ and $p.q$ are both positive real conjugate exponents with $q\geq p$ then$|A-B|^{2}+|(p-1)A+B|^{2}\leq p|A|^{2}+q|B|^{2}$.

(1)

Mathematical subject classiflcation(2000): $46B20$
Kqywords and $Phra\infty$ : Bohr inequality of operators, 3 $x.l(bIock)$ operator matrices.
The research of the work was partially supported by National Natural Science Funds of China(10726073)
and the loan of $J\iota pan$ Bank for lnternational Cooperation(JBIC).

49
It is

$\backslash \backslash Pl1$

known tbat tlie$\downarrow)_{PC,O111}p)\dagger;iti\circ l\}1=T|41|$

obtained in [2] and$[(i]$

In tliis paper, we$a1$ ) $b’o1nt\iota^{3}$. Recentl.v,

value operintor plays an$Vari_{\circ ns\prime}\cdot Jo^{1(\iota\downarrow er_{\dot{t}}\iota 1i’\text{ノ_{}d}i1tio11S}$$il[\iota po]\uparrow’(tll$of Bohr(role in$iIle(\lrcorner 11i1]it\iota^{\mathfrak{l}}es$.

$tli_{1^{J}}$$1\supset(x1_{i.1}J$

liave been.

$i_{11}\iota pro\tau^{r}\tau:’$tlie$ac\cdot cur_{\dot{t}}\iota.\iota:\backslash \cdot$of tlie estiiiiate given 1} the $ori_{S}ina1$ Bolir$)’$

As a niatter of fact, the parallelograiii law for)$i\iota 1$$s\backslash ol\iota\iota t,e$

$i11t^{1}t|ti\backslash 1it\backslash$value of $oeratorb^{\tau}$ ;(2)$|A+B|\underline,$ $+|A-B|^{\sim^{1}}=arrow?|A|^{2}+2|B|\underline,$is our$vie\backslash vpoint_{\lambda}$.

.

An operator version of the Bohr inequality (0) is obtained by a general-

ization of (2) as follows:$|A+B|^{2}+|$for operators A. $B$ , and

$r/$

}

$q\geq 1$$\Delta$; 164–$=$

with

了$B|^{2}=p|.4|^{2}+q|B|\dot$(3)$\sqrt{}\tau=$$\frac{l}{p}+\frac{1}{1l}=1$, because of $(p-1)(t/-1)=1$ .

Furthermore. we extend the Bohr inequality to a tliree variable case:If$\frac{1}{p}+\frac{1}{1l}+\frac{1}{w}=1$

for $p.q$.

$1\lambda’>0$. then for operators , B. , we have
$C$

$A$

$|A+B+C|^{2}\leq p|A|^{\underline{o}}+q|B|^{\vee}+|A^{t}|C|^{\underline{1}}$

\S 2

.

Bohr equality for 2 operators

The operator parallelogram law (2) has also the following generalizotion. which is differeiit
froni (3) a bit:Tlieorem 2.1 lf A. $B\in B(H)$ . then$|A+B| \cdot+\frac{1}{\ell}|tA-B|\underline,$$=(1+t)|A| \underline,+(1+\frac{1}{f})|B|\cdot$. for$t\neq 0$

Proof. It follows that$|A+B|^{2}+ \frac{1}{t}|tA-B|^{2}$$=$$|A|^{\tau}+|B|’-+A^{*}B+B^{*}A+t|A|^{2}+ \frac{1}{t}|B|1-A^{*}B-B^{*}A$$=$$(1+t)|.4|^{\sim^{1}}+(1+ \frac{1}{t})|B|^{\underline{Q}}$.

It is imniediately obtained from the condition of .
Corollary 2.2 (i) If $0<t\leq 1$ , then$t$$|A+B|^{\sim}+|tA-B|^{2} \leq(1+t)|A|^{\underline{r}}|+(1+\frac{1}{1})|B|_{\dot{\tau}}^{\sim^{7}}$If $t\geq 1$ or $t<0$ , then $|A+B|^{2}+|tA-B|^{\sim}\geq(1+t)|A|^{\eta}\sim+(1+1t)|B|^{\sim}\supset$ .

(ii)

As an easy consequence, we have Bohr type inequalities obtained in [2] and [4].
Corollary 2.3 [4, Theorem 1] If $A,$ $B\in B(H),$ $\frac{1}{p}+1q=1,1\leq p\leq 2$ , then(i)$|A-B|^{?}+|(p-1)A+B|^{2}\leq p|A|^{2}+q|B|^{2}$ .

(ii)$|A-B|^{2}+|A+(q-1)B|^{2}\geq p|A|^{2}+q|B|^{2}$ .

Corollary 2.4 [2, Theorem 2](iii)

If $A,$ $B\in B(H),$$\frac{1}{p}+\frac{1}{q}=1,$$|A-B|^{2}+|(p-1)A+B|^{2}\geq p|A|^{2}+q|B|^{2}$

Corollary 2.5 2, Theorem 3
$[$

$]$

If $A,$ $B\in B(H),$$p<1$. then.

$|\alpha|\geq|\beta|$, then$|A-B|^{2}+ \frac{1}{|\alpha|*}||\beta|A+|\alpha|B|^{2}\leq(1+\frac{|\beta|}{|\alpha|})|A|^{2}+(1+\frac{|\alpha|}{|\beta|})|B|^{-}$

.

50
with

$eq\iota ialit\backslash \cdot$

if and only if

\S 3

or

$|l\iota|=|\beta|$.

$|\beta|A+|\alpha|B=0_{\backslash }$Bohr-type inequalities for 3 operators

Observe that$|A+B+C|^{2}=$ $(I$.

$I$

$I)(\begin{array}{lll}|A|^{2} A^{l}B A^{*}CB^{*}A |B|^{2} B\cdot CC^{*}.4 C^{*}B |C|^{\underline{1}}\end{array})(\begin{array}{l}III\end{array})\geq 0$

Tlieii. due to the idea of [6], we may convert a probleni of absolute operntors to a problem of$3\cross 3$

block operator $nlatrices$ . while tlie later approacli maybe$ea\backslash \neg ier$

to handle.

For the sake of convenience. we cit, the following well-known fact:$e$Lemma3.1

If $x,$$y,$ $\vee-\geq 0$. and$a,$ $b,$$cER$ witli$\{\begin{array}{l}\iota\cdot\iota/\geq a^{2}, yz\geq\iota^{-}.x\approx\geq b^{\underline{\eta}};xy\approx+2\iota JtC\geq.\iota\cdot\iota^{\sim}+\iota/b^{\vee}+\approx\iota^{-}\prime’.,.,.\end{array}$Then$(\begin{array}{lll}x u bCJ y cb c \approx\end{array})\geq 0$

Lemma 3.2

Let.4

$i\in \mathbb{R}(H),$$\alpha_{i},$$\beta_{j}\in R$.

with $i=1.2,3$ . Tlien positive operator-valued

function$F(\alpha_{1}.\alpha_{2}, \alpha_{3})=|’.|1\cdot(\iota^{l}\cdot A|^{2}$is order preseiviitg if the order$(\alpha_{1},\alpha_{2}, \alpha_{3})\prec(\theta_{1}$Proof.

Siiice$/3_{J}$)

aniong$R$

is defined by$\Leftrightarrow|\alpha_{i}|\leq|\beta_{i}|$$|A_{1}+\wedge 4_{2}+A_{\delta}|^{2}=$it suffice to show that$(\begin{array}{l}\alpha_{1}A_{1}^{*}\alpha_{2}A_{2}^{*}\alpha_{3}A_{3}^{r}\end{array})$, Lil..

$\prec$$($Ifor all$I$$(\alpha_{1}, \alpha_{2}, \alpha_{3})\prec(\beta_{1}, \ , /3_{3})$$(\alpha_{1}A_{1}$$\alpha_{2}.4_{2}$

$i$

and

$\alpha_{i}\beta_{j}=\alpha_{j}\beta_{\mathfrak{i}}$$1)(\begin{array}{l}A_{1}^{*}A_{2}^{*}A_{3}^{*}\end{array})(A_{1}$$A_{2}$

for$i\neq j$.

$\mathcal{A}_{:\}})(\begin{array}{l}III\end{array})$,

implies that$\alpha_{3}A_{3})\leq(\begin{array}{l}/3_{1}A_{1}^{*}\beta_{\vee},4_{-}\beta_{?}A_{3}^{t}\end{array})(\beta_{1}A_{1}$$/3_{\wedge}\prime A\underline{\cdot}$$j3_{d}A_{3})$,that is,.

$(\begin{array}{lll}\alpha_{1}^{2}|A_{1}|^{2} \alpha_{1}\alpha_{2}A_{1}^{*}A_{2} \alpha_{1}\alpha_{3}A_{1}^{*}A_{3}\alpha_{1}\alpha_{2}A_{2}^{*}A_{1} \alpha_{2}^{2}|A_{2}|^{2} \alpha_{2}\alpha_{3}A_{2}^{*}A_{3}\alpha_{1}\alpha_{3}A_{3}^{*}A_{1} \alpha_{2}\alpha sA_{\dot{3}}A_{2} \alpha_{3}^{2}|A_{3}|^{2}\end{array})\leq(\begin{array}{lll}\beta_{1}^{2}|A_{1}|^{2} \beta_{1}\beta_{2}4_{1}^{*}A_{2} \beta_{1}\beta_{l}\prime A_{1}^{*}A_{3}\beta_{1}\beta_{2}A_{2}^{l}A_{1} \beta_{2}^{2}|\prime 4_{2}|^{2} \theta_{-},\beta_{3}A_{\sim}^{*}\circ A_{J}\beta_{1}\beta_{3}A_{3}^{*}A_{1} \beta_{2}\beta_{J}A_{3}^{*}A_{2} \beta_{\backslash !}^{2}|A_{3}|^{\wedge}\end{array})$51
By tlie (1$(sfi_{11}iti()t\iota$

nnd $Lemm:\iota 3.1$ . we lmve,$(\begin{array}{lll}\beta_{1}^{2}-tt_{1}^{2} \beta_{1}(i_{\sim^{1}}\prime-(\iota 1\alpha_{2} \beta_{1}\beta_{3}-\alpha_{1}\alpha_{3}\beta_{1}\beta_{-},--t_{1}\alpha_{2} \beta_{\sim^{1}}^{2}-tl_{2}2 j3_{2^{t}},j_{\}}-o_{\underline{1}}\alpha_{3}\beta_{1}3.-(|\cdot /^{j_{\underline{9}}\beta.\cdot.-(\iota_{-},\alpha_{3}} \beta_{4}^{2}-\alpha_{4}^{2}\end{array})\geq 0$

wliich implies the conclusion.

Theorem 3.3 Let $A,$ $B,$ $C\in B(H),$$\{\iota v\geq\gamma^{2}p\geq\alpha^{2};ll\geq\beta^{-};$$\alpha_{i}\in \mathbb{P}_{1},$$p,$$f\int$. $lU>0$ witli $i=1,2,3$ . If$\{\begin{array}{l}(l^{J}-n^{2})((1-\beta^{2})\geq(\alpha\beta)^{2};\{-f^{-})(tA_{1}^{\wedge\sim}-,,)\geq(\beta\gamma)^{2};(u-\gamma\underline{.,})(p-C.1^{-})\geq(\gamma_{C\mathfrak{i}})^{2};\mu lw\geq r_{\sim}\iota^{\underline{1}}q\downarrow v+;3^{\eta}\sim p\cdot u+\gamma^{\underline{0}}pq\prime..\end{array}$

.
Then

.

$|\alpha A+\beta B+\gamma C|^{2}\leq\rho|A|^{\underline{\backslash }}+\prime l|B|^{2}+!L^{||C|^{\sim}}.$

Proof.

As in above, we liave to sbow that$(\begin{array}{lll}\alpha^{\underline{o}}|A|^{\sim} c\nu\beta A^{*}B \cap./4;1^{*}C\iota r/3B^{*}A \beta^{2}|B|^{2} \beta\gamma B^{*}C\alpha\gamma C^{*}.4 /9\gamma,C^{*}B \gamma^{2}|C|,\cdot\end{array})\leq(\begin{array}{lll}p|\lrcorner 4|^{\underline{\eta}} 0 00 q|B|- 00 0 u\prime|C|^{2}\end{array})$.

Therefore, what we should do is just to prove that.

$(\begin{array}{lll}\rho-\alpha^{2} \alpha\beta c\iota\cdot\gamma C1/3 q-\beta- \mathfrak{l}l3\gamma(.]\gamma \beta\gamma ll’-\gamma^{\eta}\sim\end{array})\geq 0$

which is obtained by the }$\iota ssilniption$ and Lemma 3.1.
$T\mathfrak{l}\iota e$following corollary is synnnetric to Tbeorerii 3.3.

Corollary 3.4 Let A. $B,$ $C\in B(H),$$\{\iota\iota\leq\gamma^{2}p,\leq\alpha^{2};q\leq\beta^{2};$$\alpha_{i}\in \mathbb{R}$. $p.q,$ $u>0$ with $i=1.2.3$ . If$\{\begin{array}{l}(p-CV^{1}\sim)(q-\beta\underline,)\geq(Ct/f)^{2};(q-\beta^{\underline{Q}})(\iota\iota\dagger-\gamma^{2})\geq(/3\gamma)^{2};(w-\gamma^{2})(p-\alpha^{2})\geq(\wedge(\alpha)^{2};\mu l^{u\prime}\geq fJ^{2}\prime qntf+/3^{\wedge}pw+\text{ツ^{}2}pq.\end{array}$.
Then$|\alpha A+\beta B+\gamma C|^{2}\geq p|A|^{:}+q|B|^{2}+w|C|^{2}$.

Now we have Bohr inequality for 3 operators.
Corollary 3.5 If $p,$

Proof. Given$q,$$w>0,$$\frac{1}{p}+\frac{1}{q}+\frac{1}{u}=1$, then $|A+B+C|’:\leq p|A|^{2}+q|B|^{2}+lA’|C|^{2}$ .

$p,$ $q,$$w>0,$$\frac{1}{p}+\frac{1}{q}+\frac{1}{w}=1$, the following is$\{w\geq 1p\geq 1;q\geq 1;$.

$\{\begin{array}{l}(p-1)(q-1)\geq 1;(q-1)(w-1)\geq 1;(w-1)(p-1)\geq 1;pqw=qw+pw+pq.\end{array}$

52$Tli_{I^{J}}refore,$tliat$T1_{1PI)1(111,).,J}J^{\cdot}\cdot 1i_{111}pli(J\backslash$$|A+B+C^{\cdot}|^{-}\leq])|A|^{2}+q|B|^{2}+n’|C|^{2}$ .

\S 4

Bohr equality for multiple operators

For this, we begin witli the reforiiiulation of (3). As a matter of fact, it is just tlie firststep in this section:

Lemma 4.1

Let

$A_{1},$$A_{2}\in B(H)$

.

witli

$\frac{1}{\Gamma 1}+\frac{1}{r_{2}}=1$$r_{1},$ $\prime_{2}\geq 1$.

$r_{1}|_{J}1_{1}|^{2}+7_{\wedge}’|.4_{2}|^{2}-|.4_{1}+A_{-}|^{\sim}=|\sqrt{\frac{r_{1}}{\Gamma_{-}}}.4_{1}-\sqrt{\overline{r_{1}^{\sim}}\prime}A_{2}|^{2}$

Theorem 4.2

Suppose that.4; $\in B(H)$ . and$r_{i}\geq 1$

with

.

$\sum_{i=1}^{||}\perp i=1$

for $i=1,2,$$\ldots,$$r\iota,$$n\in N$

.

Theii

.

$\sum_{i=1}^{n}i_{j}|.4_{j}|^{\sim}-|\sum_{i=1}^{r1}A_{i}|^{2}=\sum_{1\leq j<j\leq r},$We sliow it by the;nductioii on

Proof.

$r\cdot\iota$

$n$

and

$\sum_{i=1^{r_{t}^{7}}}^{k}.1=1$. If we out(4). Note that it is true for $n=2$ by Lemina 4.1.

Now suppose tbat it is true for . $=A:$ . tlien we take.
$satis\mathfrak{h}\cdot ing\sum_{i=1}^{k+1},\underline{1,}=1$.

$|\sqrt{\frac{r_{i}}{r_{j}}}.4_{i}-\sqrt{\prime\prime^{\underline{j}}i}.4_{j}|’\sim$$\prime_{\acute{j}}=1_{l}(1-\frac{1}{r_{k+1}})$$4_{1}$.

$\cdots,$

for $i=1,$$A_{k+1}\in B|H)$ and $r_{1},$ $\cdots,$ $r_{k+1}>1$$\cdots,$$A$: for convenience, then$l_{\acute{i}}>1$. Herice we liave
な十 1

$k+1$

ゐみ

$\sum_{i=1}r_{i}|A_{i}|^{2}-|\sum_{j=1}\mathcal{A}_{;}|^{2}=\sum_{i=1}r_{l}|A_{t}|\underline{.,}+r_{k+1}|A_{k+1}|^{2}-|\sum_{i=1}A_{j}+A_{k+1}|^{\underline{\prime)}}$$=$$(1- \frac{1}{r_{k+1}})\sum_{i=1}^{k}r_{i}|.4_{j}|\begin{array}{ll}.

- -\end{array}| \sum_{i-1}^{k}A_{l}|^{2}$$+(r_{k+1}-1)|A_{k+1}|^{2}+ \frac{1}{r_{k+1}}\sum_{i=1}^{k}r_{i}|A.|^{2}-(\sum_{i=1}^{A}A_{i})^{*}.4_{k+1}-4_{k\cdot\neq 1}^{*}\sum_{i=1}^{k}A_{j}$$=$$( \sum_{i=1}^{k}r_{i}^{l}|A_{i}|\begin{array}{ll}) - -\end{array}| \sum_{i=1}^{k}A_{i}|^{2})+\sum_{i=1}^{k}\frac{r_{i}}{r_{k+l}}|A_{i}|^{2}-(\sum_{i=1}^{k}A_{i}){}^{t}A_{k+1}-.4_{k+1}^{*}\sum_{i=1}^{\lambda}A_{i}+(r_{k+1}-1)|.4_{k+1}|^{2}$$=$$1 \leq$\sum$i<j\leq k|\sqrt{\frac{r_{i}}{r_{j}}}A_{j}-\sqrt{\frac{r_{j}}{r_{i}}}A_{j}|^{2}+\sum_{i-1}^{k}\frac{r_{i}}{r_{k\cdot+1}}|A_{j}|^{2}-(\sum_{i=1}^{k}A_{i})^{*}A_{k+1}-A_{k+1}^{*}\sum_{i=1}^{k}.4_{i}+\sum_{i-1}^{k}\frac{r_{k+1}}{r_{i}}|.4_{k+1}|^{2}$$=$$\sum_{1\leq i<j\leq k+1}|\sqrt{\frac{r_{1}}{r_{j}}}A_{2}-\sqrt{\frac{r_{j}}{r_{i}}}A_{j}|^{2}$.

Therefore, the equality (4) holds for all $n\in N$ .

Corollary 4.3 [6, Theorem 7]$i=1,2,$$\ldots,$$n$

Suppose that $A_{i}\in B(H)$ , and $r:\geq 1$ with$\sum_{i=1}^{n}r_{(}\perp=1$

for. Then$| \sum_{i=1}^{n}A_{1}|^{2}\leq\sum_{i=1}^{n}r_{i}|A_{i}|^{2}$.

Equivalently, we can say that $K(z)=|z|^{2}$ satisfies (operator) Jensen inequality, in thesense that
$K( \sum_{:=1}^{n}t_{i}A_{i})\leq\sum_{i-1}^{\mathfrak{n}}t_{i}K(A:)$

53
for

, $t_{f},$$t_{1}.\cdots$$\geq 0\backslash \backslash \prime ith\sum_{?=1}^{n}t_{l}=1$

Corollary 4.4$Let_{I}\lrcorner t,$,$\in B\ovalbox{\tt\small REJECT} ff).\sum_{=j1}^{f1}\frac{1}{1i}=1,\dot{i}\dagger$md$r_{i}\neq|)$

vvith$\sum_{=1}^{\prime\iota}\frac{1}{ri}=1$

for $i=1,2,$$\ldots,$$n,$$\gamma|\in l^{\backslash }\backslash 1$

Then$\sum_{i=1}^{n}\prime_{j}|44_{i}|^{\wedge}-|\sum_{j=1}^{\prime 1}A_{i}|\underline{)},=\sum_{\iota\leq i\underline{\text{く}}j\leq f\prime}\frac{r_{j}}{r_{j}}|^{\underline{\prime_{i}}},_{j}.\cdot A_{i}-A_{j}|^{2}$\S 5
Theorem 5.1.

HUrther generalization of Bohr inequality

Let $A,$ $B,$ $C\in B\langle H)$ .

$\{\neg/^{_{\underline{Q}}}\geq_{1t1’}1+c\iota_{\underline{\tilde{0}}}^{1}:\wedge/1\geq 1+\alpha_{\hat{1}};/\cdot;;\geq+\frac{\eta}{:;}$$0_{i}\in \mathbb{P}_{\iota}$.

$\gamma_{i}>0$

witli $i=1,2.3$ . If$\{\begin{array}{l}[\gamma_{1}\prime-(I+0_{\tilde{1}})]\dot{[|}’!^{-(1+)}\alpha^{\sim^{1}}.)]\geq(1+a_{1^{tX\prime}\underline{?}})\underline{.,}:[\hat{l}^{\underline{\tau}-(1+ry_{\overline{\underline{\eta}}})][\cdot-(1+\alpha_{\hat{:)}})]}\prime’ j:;’\geq(1+\alpha_{2}rx_{J})^{z_{:}}[\gamma]-(1+\alpha\dot{]})][\gamma;\-(1+c\iota_{\tilde{J}})]\geq(1\iota v_{1}\alpha)-.\end{array}$.
with$[\gamma_{1}-(1+C_{-}V^{\frac{}{1})]\dot{[}-}\mathfrak{l}2(1+\alpha_{\dot{2}}^{\eta})][\gamma_{3}-(1+\alpha_{?}^{\circ}\cdot)]-2(1+\alpha_{1}a_{2}’)(1+(-\gamma_{1:\}}\nu)\langle 1+n_{z:J}\alpha)\geq$$-[_{\hat{i}S}-(1+u_{\tilde{J}}^{\backslash })](0_{12}’\wedge;^{\underline{\eta}}-(1+c\iota_{\underline{\tilde{0}}}^{Q})](1+\iota\iota_{1}^{l}\epsilon\iota_{S}^{\iota})^{2}-[1+^{z}\cdot,.$.

Then$|A_{1}+A_{\underline{n}}+A_{;:}|^{\sim}+|\alpha_{1\wedge}4_{1}+\iota_{2}^{r}.4_{2}+a_{\theta}A;;|\underline{.,}\leq\gamma_{1}|.4_{1}|^{-}+\wedge/:|A\cdot.|\underline{.,}+\gamma_{?}^{l}|A:f|^{\tau}\sim$

Proof.

Notice that both sides of the inequality (5) correspond to$(\begin{array}{lll}(\iota\prime (l+\alpha_{1}\alpha_{2})A_{1}^{*}A_{-} (l+\alpha_{1}\alpha_{3})A_{1}^{*}4_{3}\{l\prime\iota v_{\wedge})A_{\eta}^{*}.A_{1} (1+(l_{-}^{\S})|A_{\sim}|^{\sim} (Ct_{*}^{r}\}\cdot\cdot(1+a_{1}r_{\prime}r_{\delta})A_{\delta}^{*}A_{1} (1+\alpha_{\underline{7}}\alpha_{3}]A_{l}^{*}.4_{-} (1+\alpha^{\frac{\supset}{‘ t}})|A_{J}|^{\dot{\gamma}}\end{array})$aud$(\begin{array}{lll}\gamma_{1}|.4_{1}|^{\underline{o}} 0 00 \gamma_{2}|A_{1}|^{\underline{n}} 00 0 \gamma_{?}|A_{t}|^{2}\end{array})$

respectively. Hence, it is suffice to show that.

$(\begin{array}{llllll}-\gamma](l+ \alpha^{\frac{Q}{1}}) -l-rx_{1^{CJ’\underline{1}}}\cdot -l-\alpha_{1}\alpha_{3} -1-a_{1}c\backslash z -\gamma_{\underline{1}}(l+t_{\prime}Y_{\underline{1}}^{1}) -1- \alpha_{-Ct_{\theta}^{}}-l- \alpha_{1}\alpha_{3} -1- -\alpha\cdot\alpha_{3}(l+ \gamma_{\delta}\prime a_{d}^{4})\end{array})\geq 0$

which is implied by the assumption and Lemma 3.1.
Corollary 5.2 Let A. $B,$ $C\in B(H),$$\alpha_{1},\alpha_{2}.\beta_{i}\in \mathbb{R}$.

$\gamma_{i}>0$with $i=1,2.3$ . If;$\gamma_{2}\gamma_{1}\geq 1\geq 1I_{\alpha_{\tilde{2}}^{\eta};}^{\alpha^{o}}1$$\{$$\{\begin{array}{l}[\gamma_{2}-(\alpha_{2}^{2}\prime+1)](\gamma_{3}-1)\geq 1;[\gamma_{1}-(\alpha_{1}^{2}+1)](\gamma_{3}-1)\geq 1:[\gamma_{1}-(\alpha_{1}^{2}+1)][\gamma_{2}-(\alpha_{2}^{2}+1)]\geq(1+\alpha_{1}\alpha_{\underline{0}}):_{;}1+\alpha_{1}\alpha_{2}\leq 0.\end{array}$$\gamma_{3}\geq 1$

.

Then

.

$|A_{1}+A_{2}+A_{3}|^{2}+|\alpha_{1}A_{1}+\alpha_{2}A_{2}|^{2}\leq\gamma_{1}|A_{1}|^{2}+\gamma_{2}|A_{2}|^{2}+\gamma_{3}|A_{3}|^{2}$

.

(5)

.

54
of

$\backslash 1\downarrow\downarrow)t1.\iota^{J}\iota l_{\llcorner}\backslash$Corollary 5.3

iiieqiialitv is presei $lted$ :$B()111^{\cdot}$

lf.4. $B,$ $C”\in B|H|$ . and$\frac{1}{\int)}+\frac{1}{1l}+\frac{1}{w}=1,$]$J.(/\cdot(t’\geq 0$, theii$|.4+B+C|^{\sim^{1}}+| \frac{\rho}{\vee\mp}.4-\frac{(l}{\sqrt{l^{J+}l}}B|\underline{.)}\leq\rho|.1|^{:}+q|B|^{\sim^{1}}+11’|C^{\cdot}|^{\underline{\iota}}$

Theorem 5.4

Let.$4_{j}\in B(H),$ $0_{i}.’;\in R$ .

$\{\wedge/3\geq\overline{/}n\hat{l}1_{\geq}\sim\geq 1+f1^{\frac{o}{:\}}}1+^{Q}1+(- t^{\mathring{\frac{}{1}}};CJ_{\sim}’\sim;$$\hat{|}i>1)$.

witli $i=1,2.:;$ . If$\{\begin{array}{l}[\gamma_{1}-(r.\iota_{\overline{1}}+1)][\gamma_{2}-(\alpha^{\frac{\backslash }{\cdot\cdot)\sim}}+1)]\geq|\alpha_{1}(\lambda_{2}+1]^{2};[\gamma_{-}’-\langle\alpha^{\sim}\underline{.}+1)][\gamma;\}-(\iota r_{\tilde{J}}+1]]\geq t\alpha_{2}\alpha_{\theta}+1)^{!_{;}}\alpha_{1}(t::+1=0.\end{array}$Then$|.4_{1}+.4_{-}+A_{S}|^{2}+|+:\}\cdot:\}.1\leq\gamma_{1}|.4_{1}|^{2}+\wedge j\underline{\mathfrak{n}}|.4_{-}|^{2}+\gamma_{?}|.4_{3}’|^{2}$

Corollary 5.5

If A. B. $C‘\in B(H)$ , and$\frac{1}{p}+\frac{1}{l\int}+\frac{1}{1}=1.p.q$.

$\iota\iota’\geq 0$.

. then$|A+B+C|^{\underline{\eta}}+| \frac{1}{\sqrt{u|-1}}.4+\frac{1}{\sqrt{u|-1}}B-\sqrt{|\iota\prime-1}C|^{2}\leq\rho|.4|’-+q|B|^{!}+\dagger A’|C|^{2}$.

Related to [3]. we have the following inequalities. As a inatter of fact. the right-hand sides. $|B|-$ and $|C|-$ in [3. Lemiiia 1],
If.4. B. $C\in B(H)$ . and $t\in(O. 1)$ . tben we haveare regarded as tbe weighted ar\’ithmetic meati ofCorollary 5.6$|.4|^{2}$$|A+B+C|^{2}+| \sqrt{t}A+\sqrt{t}B-\frac{1}{\sqrt{t}}C|\underline{.,}\leq\frac{\underline{?}_{-}t}{1-t}\frac{1+t}{t}|A|^{2}+\frac{2-t}{1-t}|B|^{2}+\frac{1+t}{t}|C’|^{2}$:.

$|A+B+C’|^{2}+| \sqrt{]-t}(A+B)-\frac{1}{\sqrt{1-t}}C|^{-}\leq\frac{1+t}{t}|.4|^{-}+\frac{\underline{9}_{-}1}{1-t}\frac{1+t}{t}|B|^{2}+\frac{2-t}{1-t}|C|^{2}’$Acknowledgement

Tbe authors would like to express tlieir thanks to Professor J.1. Fujii

for intro(luciiig tlie reference [2].

References[1] Zur theorie der fastperiodischen funktionmen I, Acta Math., 45(1927), $29\cdot 127$ .

[2] W.-S. Cheung and J.Pecaric, Bohr’s inequalities for Hilbert sapce operators, J.

Math. Anal. Appl.. 323(2006), 403-412.

. Fujii. M. Nakaniura, J. Pecaric and . Seo, A reverse inequality for
the weighted geometric mean due to Lawson-Lim, Linear Algebra Appl., 427(2007),.

[3] J. I. Fujii,$\backslash \downarrow 1$$Y^{\cdot}$$2\overline{/}2- 284$

[4] O. Hirzallah, Non-commutative operator Bohr inequalities, J. Math. Anal. Appl.,
282(2003),$5\overline{/}8- 583$.

[5] N.Young, An Introduction to Hilbert Space, Cambridge University Press, 1988.
[6] F. Zhang, On the Bohr inequality of operators, J. Math. Anal. Appl., 333(2007),1264-1271.