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On Powers of p-Hyponormal and Log-Hyponormal Operators

TAKAYUKI FURUTA andMASAHIRO YANAGIDA*

Departmentof AppfiedMathematics,Faculty ofScience, ScienceUniversity ofTokyo, 1-3 Kagurazaka, Shinjuku, Tokyo 162-8601,Japan

Dedicatedtothememoryof Prof. Sz6kefalvi-Nagy,B61aindeepsorrow (Received17May1999;Revised14 July1999)

Aboundedlinearoperator Tona HilbertspaceHis said tobep-hyponormal forp>0if

(T* T)^p>(TT*)P,^and Tis said tobelog-hyponormalifTis invertibleand log T*T>

logTT*.Firstly,weshall show the followingextension of ourprevious result: IfTis p-

(+l)/n> > ^{(p+l)/2 p+l}

hyponormal for pE(0,1], then (T’T) _"’_(T’T) >(T T) and

P+] (p+l)/2 (p+l)/n

TT > T T > > T T*) hold for all positiveintegern.Secondly, weshalldiscuss the bestpossibilities of the following parallel result for

log-hponormal

operators by Yamazaki: If T is log-hyponormal, then (T^n,Tn)/n^>...>

1/2 1/2 l/n

(T’T) ^>r r and TT >(r T*) >...> (T T*) hold for all

ffositiv

integern.

Keywords." p-Hyponormal operator; Log-hyponormal operator;Furutainequality 1991 MathematicsSubjectClassification: ^Primary47B20; 47A63

1

INTRODUCTION

A

capital lettermeans abounded linear operatoron acomplexHilbert space H.

An

operator Tis said to be positive

(denoted

by T>

0)

^if

(Tx, x)>_

0 for all x

EH

^{and also an} operator Tis said to be strictly positive

(denoted

^{by T}

> 0)

ifTispositiveandinvertible.

* Correspondingauthor. E-mail:[email protected].

E-mail:[email protected].

367

An

operatorTis said tobep-hyponormal forp

>

Oif(T*T)^p

>_ (TT*)

^p andanoperator Tis said to be log-hyponormalifTisinvertible and logT*T_> logTT*.p-Hyponormal andlog-hyponormal operators are defined as extensions of hyponormal one, i.e.,

T’T>_

TT*. It is easily obtained that every p-hyponormal operator is q-hyponormal for

p._>

q

>

0 by the celebrated L6wner-Heinz theorem "A

_>

B

>

0 ensuresA

>

B

for

^anyaE

^[0,

^]," and every invertiblep-hyponormal operatorislog-hyponormalsincelog ^{is an}operatormonotonefunction.

We remark that

(A

^{p-1)/p log A as} p +0 for positive ^invertible operator A

>

0, so that p-hyponormality of T approaches log-hypo- normality of T as p+0. In ^this sense, log-hyponormal ^{can be} consideredas0-hyponormal.

Recently,Aluthgeand

Wang [2]

showed the following results.

THEOREM

A [2]

^LetTbeap-hyponormaloperator

for

^pE

^{(O, 1].}

^Then

(r

ⁿ

rn)

^p/n

>_ (T’T)

^p

>_ (TT*)

^p

^>_ (Tnrn*)

^p/n

(1.1)

hoM,thatis,

T"

is(p/n)-hyponormal

for

^allpositive integer^n.

Itiswell known thatevenifTishyponormal, T^{2is not}hyponormal in general

[9,

Problem

209],

but paranormal

[4],

i.e.,

[[T2x[[ ^_> Tx[[

²

holds for everyunit vector x.Nowit turns outbyTheorem

A

thatT²is (1/2)-hyponormal for every hyponormal operator T, which is more precisesince(1/2)-hyponormalityensuresparanormality

[1,7].

Very

recently,in

[8],

^{weshowedanextensionof Theorem}

A

asfollows.

THEOREM B Let Tbeap-hyponormal_operator

for

^p

^{(0, 1].}

^Then

(Tn*Tn)

^(p+l)/n

^>_ (T’T)

^p+I

(1.2)

and

(TT*)

^p+I

^>_ (TnTn*)

^(p+)/n

hold

for

^allpositive integern.

(1.3)

Wealsodiscussedthe best possibilities of Theorem and Theorem

A.

On the other hand, ^Yamazaki

[12]

showed another extension of Theorem

A

asfollows.

THEOREM C

[12]

LetTbeap-hyponormaloperatorforp

(0, 1].

Then

(T

^n*

Tn)

^1/n

_ ^{>_ (T}

^2.

^T2)

^1/2

_

^T*T

^(1.4)

and

TT* (T2T2*)

^1/2

... ^(TnTn*)

^1In

^(1.5)

hold

for

^allpositive integer n.

Weremark that Theorem

A

follows from TheoremB

(or

^Theorem

C)

obviously.In fact,the first andthirdinequalitiesof

(1.1)

^holdby

(1.2)

and

(1.3)

^{of Theorem}B

(or (1.4)

^and

(1.5)

^{of Theorem}

C)

and L6wner- Heinz theorem, and the second inequality of

(1.1)

holds since T is p-hyponormal.

Yamazaki

[12]

alsoshowed thefollowingTheoremDandCorollary E forlog-hyponormaloperators whichareparallelresultstoTheoremC and Theorem

A

forp-hyponormaloperators,respectively.

THEOREM D

[12]

LetTbealog-hyponormaloperator. Then

(Tn*Tn)

^1In

(r2*r2)

^1/2

^r*r (1.6)

and

TT* >_ (T2T2*)

^1/2

>_... >_ (TnTn*)

^1/n

hold

for

^allpositive integern.

COP,OLLARYE

[12]

LetTbealog-hyponormaloperator. Then

log(Tn*Tn)

^1/n

>_

log

T’T>_

log

TT* >_ log(TnTn*)

^1/n

hold,thatis,

T"

^isalsolog-hyponormal

for

^allpositive integer n.

(1.7)

(1.8)

We remark that Corollary E is more general than the following result by Aluthge and

Wang [1] "If

^Tis log-hyponormal, then

T2"is

log-hyponormal

for

^anypositive integern."

In this paper, we shall show Theorem stated below which is an extensionofboth TheoremBand TheoremC. Weshall also discussthe best possibilities of TheoremDandCorollary E.

2

AN EXTENSION OF BOTH THEOREM B AND THEOREM C

THEOREM Let Tbeap-hyponormaloperator

for

^pE

^{(0, 1].}

^Then

(T

^n*

Tn)(p+l)/n >_... >_ (T

^2.

T2)(p+I)/2 >_ (T’T)

^p+I

(2.1)

and

(TT*)

^p+l

^>_ (T2T2*)

^(p+1)/2

^{>_... >_} (T"Tn*)

(p+l)/"

(2.2)

hold

for

all /)ositive^integern.

Weremark that TheoremBfollows fromTheorem bycomparing the first and last terms of each of the inequalities, and Theorem C also follows from Theorem 1 byL6wner-Heinz theorem.Itisinterestingto remark that TheoremDjustcorrespondstoTheorem in_case/) 0since log-hyponormal^can be considered as 0-hyponormal^asmentioned in Section1.

InordertogiveaproofofTheorem 1,^weuse the following TheoremF.

THEOREM F

(Furuta

inequality

[5]) If

^A

^>_

^B

^>_

^O,then

for

^eachr

^>_

^O,

(i)

(Br/2APBr/2)

^1/q

^>_ (Br/2BPBr/2)

^1/qand

(ii)

(Ar/2APAr/2)I/q _ (Ar/2BPBr/2)

^1/q

hold

for

^p

^>_

^{0and q}

^>

^with

⁽¹ + ^r)

^q

^>_

^p

+

^r.

Weremark that TheoremFyieldsL6wner-Heinztheorem when we putr 0in(i)^or(ii)stated above.AlternativeproofsofTheoremFare givenin

[3,10]

and alsoanelementaryone-page proofin

[6].

Itisshown in

[11]

that the domain drawn for p, q andrinFig. isthe best possible for TheoremF.

(0, -r)

(1 + ^{r)q p} +

^r

(1,1)

(1,0)

FIGURE

Wealsousethe following result^whichisanapplication of TheoremF.

THEOREM

F’ If

^A

^>_

^B

^>_

^0,then thefollowingassertionshold."

(i)

for

^{each q}

^>_

^{0 andr}

>_ O, f(s)= (Br/)ASBr/2)(q+r)/(

^{+r) is} increasing fors>_q.

(ii) for

^{each q}

^>_

^{0 andr}

^{>_ O,}

^g(s)

(Ar/ZBSAr/2)

^{(q+r)/(s+r) is decreasing}

fors>_q.

Proof of

^{Theorem 1 Let T=}

UITI

^{be the}polar decompositionof T.

Thenit iswellknown that thepolardecomposition of T*isT*

U*I ^T*[.

Put

An=(Tn*Tn)p/n= _[Tn[

^2p/nand

Bn=(TnTn*)p/n= IT"*[

^{2p/" for each}

positiveintegern.

Proof of ^{(2.1) We}

^{shall prove that the following}

^(2.3)

holds for all positive integer n,which isequivalentto

(2.1)

obviously:

(Tn+l*Tn+l)(p+l)/(n+l) (T

^n*

Tn) ^(p+l)/n. (2.3)

(i) Firstly,weprovethat

(2.3)

^{holds forn 1,that is,}

(T

^2.

T2)(p+I)/: >_ (T’T)

^p+I

(2.4) A1--(T*T)P>_(TT*)P=B1

holds since T is p-hyponormal.

By

applying (i)of TheoremFto

A1

^and

B

^forlip

^{>_ O,}

^wehave

(T

^2.

T2)

^(p+l)/2

U*([T*[T*TIT*I)(P+I)/2U

U*(BI/2pAI/PBI/2p)(I+I/P)/(1/p+I/p) u

U’B]

^+l/pU

U*IT*I2(p+I)

U

IT[

^2(p+l)

(T’T) p+I,

sothat

(2.4)

^isproved.

(ii) Secondly,ⁱⁿordertoprove that

(2.3)

holds forn

_>

2,weprovethe following

(2.5)

^byinduction:

(T

^n+l*

Tn+l)

^{n/(n+l) T}

^{’* T"}

for all positive integer n.

(2.5)

We remark that

(2.5)

^impliesthat

(2.3)

^{holds forn}

>

2 byapplying

L6wner-Heinz

theoremto

(2.5)

for(p

+ 1)/n

^E

(0, 1].

(2.5)

^{holds forn by}

(2.4)

^andL6wner-Heinztheorem.

Assume

that

(2.5)

^{holds for}n-1,2,...,k- 1.

By

applyingL6wner-Heinztheorem to

(2.5)

forp/nE

(0,

1],^wehave

(T

^+*

Tn+l)

^p/(n+l)

_ ^{(T * Tn) p/n,}

^sothat

Ak (T

^k*

rk)

^p/k

_ ^(r

^2.

^T2)

^p/2

^{>_ (T’T)}

^p

^{>_ (TT*)}

^p

^B.

The last inequality holds since T is p-hyponormal. Put ql

=(k- 1)/

p>0 and rl=l/p>O. Then by (i) ^{of Theorem}

F’, f(s)--

ictr/2As .or,/2 (q,+r)/(s+r,) lI/2pAs.ol/2p

"1 ’k’l --k"l ’k’l

)k/(ps+l)

is increasing for

s

>_

q

(k

1)/p,sothatwehave

T^k*T

k.

The last inequality holdssince we assumethat

(2.5)

^{holds for}n k-1.

Hence

(2.5)

also holds forn-k,^sothatit isprovedthat

(2.5)

^{holds for} all positive integern.

Consequently,theproofof

(2.1)

^iscomplete bycombining(i)and(ii).

Proof of ^(2.2)

^{We shall prove} that the following

(2.6)

holds for all positive integer n, whichisequivalentto

(2.2)

^obviously:

(TnTn*)

^(p+l)/n

(Tn+lTn+l*) (p+l)/(n+l). (2.6)

(i) ^Firstly,weprovethat

(2.6)

^{holds forn 1, thatis,}

(TT*) + _{>_} (T:T*) ^(+1)/2. (2.7) AI-(T*T)P>_(TT*)P--B1

holds since T is p-hyponormal.

By

applying (ii)ofTheoremFto

A

^and

B

^for1/p

^>

^0,wehave

(T2T2*)(P+l)/2= (UITITT*ITIU*)(P+/

U(ITITT*ITI)(P+I/U

U(A1/2pR1/pA1/2p)(I+I/P)/(1/p+I/P)

_"1 U*

I+I/Pu,

<<_

UA

UIT[

^2(p+1)U*

--[T*[2(p

⁺¹⁾

(TT*)P

⁺¹

sothat

(2.7)

^isproved.

(ii) Secondly, in order to prove that

(2.6)

^{holds for n}

_>

2, we prove the following

(2.8)

by induction:

TnT

^n*

_ (Tn+lTn+l*)n/(n+l)

^forall positive integer n.

(2.8)

We remark that

(2.8)

implies that

(2.6)

^{holds for n}

>

2 by applying L6wner-Heinztheoremto

(2.8)

^for(p

+ ^1)/n

^E

^{(0, 1].}

(2.8)

^{holds forn} by

(2.7)

^andL6wner-Heinztheorem.

Assume

that

(2.8)

^{holds forn--1,2,...,k- 1.}

By

applyingL6wner-Heinz theorem to

(2.8)

forp/nE

(0,

1],^wehave

(TnTn*)

^p/n

^>_ (T

^n+l

Tn+l*)p/(n+l),

^sothat

A1 (T’T)

^p

^>_ (TT*)

^p

^>_ (TT2*)

^ply >_""

>_ (TITk*)

^p/I

BI.

The first inequality holds since Tisp-hyponormal. Put ql--(k-

1)/

p>0 and rl=l/p>O. Then by (ii) of Theorem

F’, g(s)-

(Al/2BcAl/2)(q’+ri)/(s+r’)--(A1/2pRsA1/2p)k/(ps+l)--

"k"l is decreasing for s

>_

ql

(k

1)/p,^sothatwehave

T^kT

k"

The last inequality holdssince we assumethat

(2.8)

^{holds forn--k- 1.}

Hence

(2.8)

alsoholds forn k,sothatit isprovedthat

(2.8)

holds for all positive integern.

Consequently,theproofof

(2.2)

^iscompleteby combining(i)and(ii).

BEST POSSIBILITIES OF THEOREM D AND COROLLARY E

The following Theorem2 assertsthe best possibility of TheoremD.

THEOREM 2 Letn

>_

2 and

>

^1.The thefollowing hold:

(i) there exists a log-hyponormal operator Tsuch that

(Tn*Tn)/n

(T’T) .

(ii) there exists a log-hyponormal operator T such that

(TT*)_

(TnTn*)d n.

Weremark thatA

>

B for6

>

⁰approaches log A

>

log Bas6 +0 for positive invertible operatorsA and B.Inthis sense, thefollowing Theorem 3 asserts the best possibilities of all the inequalities of

(1.8)

inCorollaryE.

THEOREM 3 Letn

>_

anda

>

^O. Then thefollowing hold."

(i) there exists a log-hyponormaloperator Tsuch that

(Tn*T")/n_

(ii)

there exists a log-hyponormaloperator Tsuch that

(T"*T")/"_

( ,).

(iii)

there exists a log-hyponormal operator T such that

(T*T)_

(TnTn*)cel n.

Togiveproofs of Theorem 2 and Theorem 3, we use the following results.

PROPOSITION

[13]

Letp

>

0, q

>

0 andr

>

^O.

If

^rq

^<

^p

+

^{r, then the}

followingassertionshold."

(i) ^{there exist}positive invertible operators A andB on I² such that log A

>

log Band

(Brl2APBr/2)

^{1/q B(p+r)/q.}

(ii) there exbct positive invertible operators A andB on

]2

_{such that} log A

_>

logBand

A^(p+r)lq

(Arl2BpArl2) ^llq.

LEMMA Forpositive operatorsAandBon

H, define

^{theoperatorTon}

(R)k_oH

^as

follows."

0

Bll

² 0

Bll

²

.41/2

₀

All

² 0

(3.1)

where^[-qshows theplace

of

^the

^{(0, O)}

^matrixelement. Thenthefollow&g

assertionshold:

(i) T^isp-hyponormal

for

^p

^>

⁰

if

^andonly

if

^Ap

^>_

^B

^p.

(ii) T ^is log-hyponormal

if

^{and only}

if

^{A and B are invertible and}

log A

_>

log B.

Furthermore,thefollowing^assertionshold

for ^>

^{0 andintegers n}

^>_

^2:

(iii)

(T

^n*T

n)

^/n

>_ (T, T) if

^andonly

if

(Bk/2An-kBk/2)e/n ^>_

^B

^e

^holds

for

^{k 1,2,...,n- 1.}

(iv)

(TT*) >_ (T"T"*)

^/n

_if

andonly

if

A

e _{>_} (Ak/VBn-kAk/)e/n

^holds

for

^{k 1,2,...,n 1.}

(v) (Tn*Tn)

^/n

^>_ (TnTn*)

^/n

if

^andonly

if

A

>_

B

e

_{holds and}

(Bk/2An-kBk/2)

^e/n

^>_

^B

^e

A

e _{>_} (A/Bn-gAk/2)

^/n

and

hold

for

^{k 1,2,...,n-1.}

(3.2)

(3.3)

(3.4) Proof ^By

easy calculation,wehave

B B T*T-

and

TT*

B

A

A A

A .j

sothat(i) and(ii)^areobvious bycomparing thetwo

(0, 0)

elements of T*T and TT*.Furthermore,thefollowinghold forn

>

2:

B(n- ^/2AB(n-O/

BIA.-kBkI

B/ZAn-IB1/

and

T Tⁿ /.

B

AI/2Bn-IB1/2

Ak/2Bn-kAk/2

A(n-I)/2BA(n-1)/2 A

A

so that we have (iii), (iv) and

(v)

by comparing the corresponding elements of

T"*T"

and

TnT n*.

Proof of

^{Theorem 2 Put}Pl--n-

> O,

ql---n/a

>

0 and rl--

> O,

thenwehaverlql

n/a >

ⁿ=pl

+ r.

378 T.FURUTA AND M. YANAGIDA

Proof of

⁽ⁱ⁾

^By

⁽ⁱ⁾ of Proposition 1, there exist positive invertible operatorsAandBonHsuch that

log.4

_>

log B

(3.5)

and

(Br/2A

^pB

r/2)

^1/q

^B(Pl+r)/q,

^{that is,}

(B1/2An-lB1/2)a/n _

^B

. ^(3.6)

DefineanoperatorTon

(R)k=_Has (3.1).

^{Then Tis}log-hyponormal by

(3.5)

and(ii)ofLemma 1, and

(T"*T")

^/"

_ ^(T’T)

^{by (iii)ofLemma}

sincethecasek of

(3.2)

^doesnothold

for/3

^aby

(3.6).

Proof of

⁽ⁱⁱ⁾

^By

⁽ⁱⁱ⁾ of Proposition 1, there exist positiveinvertible operatorsAandBonHsuch that

log A

>

log B

(3.7)

andA^(p+r’)/q

_ (Ar/2BP’Ar,/2)

^1/q’ that is,

A

(A1/2Bn-IA1/2) ^/n. (3.8)

Define anoperatorTon

@k=_Has (3.1).

ThenTislog-hyponormalby

(3.7)

^and(ii) ^ofLemma 1, and

(TT*) _ (T"T’*)

^/"by (iv)^ofLemma sincethecasek of

(3.3)

doesnothold

for/3

^cby

(3.8).

Proof of

^{Theorem 3}

Proof of

^{(i) It is well} known that there exist positive ^invertible operatorsAandBonHsuchthat

log A

_>

log B

(3.9)

and

A_B ’. (3.10)

Define an operator Ton

(R)=_H

^as

^(3.1).

^{Then Tis}log-hyponormal by

(3.9)

and

(ii)

^ofLemma1, and

(T"* T")

^/"

_ ^(TnTn*)

^a/nforn

^_>

^2by

^(v)

ofLemma since thefirstinequality of

(3.4)

^doesnothold

for/3

^c by

(3.10),

^and

(T’T) (TT*)

by

(3.10)

and(i)ofLemma1.

Proof of

^{(ii) We haveonly to provethe case}

^>

^a

^>

^{0 by L6wner-}

Heinztheorem.

Assume

(T

ⁿ

Tn)

^c/n

>_ (TT*) . ^(3.11)

Then we have

(T

ⁿ

Tn)a/n (TT*) (TnTn*) ^/n.

The firstinequalityis

(3.11)

itself, and the second inequalityholds by

(1.7)

^inTheoremDand L6wner-Heinz theorem.Thisis a contradiction to(i)of Theorem 3.

Proof of

^{(iii) Wehaveonlyto provethecase 1}

^>

^a

^>

^{0 byL6wner-}

Heinztheorem.

Assume

(T’T) ^>_ (TnTn*) ^a/n. (3.12)

Thenwehave

(Tn*Tn)

^/n

^>_ (T’T) ’ ^>_ (TnTn*) ^a/n.

The firstinequality holds by

(1.6)

^{in Theorem D and} L6wner-Heinz theorem,and thesecond inequality^is

(3.12)

itself. This is a contradiction to(i)of Theorem 3.

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