Generalizations of the Results on Powers ofp-Hyponormal Operators

J. ofInequal.&Appl., 2001, Vol. 6,pp.1-15 Reprints available directlyfrom the publisher Photocopying permitted bylicenseonly

(C)2001OPA(OverseasPublishersAssociation) N.V.

Publishedbylicenseunder the Gordonand Breach Science Publishersimprint.

Printed inSingapore.

Generalizations of the Results on Powers ofp-Hyponormal Operators

MASATOSHI ITO*

Departmentof Appfied Mathematics, Faculty ofScience, ScienceUniversity of Tokyo, 1-3 Kagurazaka, Shinjuku, Tokyo 162-8601,Japan

(Received31August1999; Revised 10September1999)

Recently,as aniceapplication ofFurutainequality,Aluthge andWang (J.Inequal. Appl., 3(1999), 279-284)showed that"/f^{Tis a}p-hyponorma!operatorfor^pE(O,1], then T isp/n-hyponormalfor ^anypositive integer n," and Furuta and Yanagida (Scientiae Mathematicae, toappear) proved themoreprecise resultonpowers ofp-hyponormal operatorsforpE(0,1]. Inthispaper, moregenerally, by usingFurutainequalityrepeatedly, weshall show that "/f^Tisap-hyponormaloperatorfor^p>O, then T ismin{1,p/n}- hyponormalfor anypositive integern"andageneralization of the resultsbyFurutaand Yanagidain(Scientiae Mathematicae,toappear)onpowers ofp-hyponormaloperators forp>0.

Keywords: p-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) >

0forallx EH.

An

operator Tis saidtobep-hyponormal forp

>

0if(T*

T)P> (TT*)p.

p-Hyponormal operatorsweredefined as anextension ofhyponormal ones, i.e., T’T>TT*. It is easily obtained that every p-hyponormal operator is q-hyponormal for p

>

q

>

0 by L6wner-Heinz theorem

"A >

B

>

0ensuresA

> B for

^{anya [0,}

^1],"

^{andit is}well known that

* E-mail:[email protected].

thereexists ahyponormal operator Tsuch that T²isnothyponormal

[13],

butparanormal

[7],

i.e.,

T2xll ^{> Txl[}

²for every unitvectorxEH.We remark thateveryp-hyponormal operator forp

>

0 isparanormal

[3] (see

also

1,5,10]).

Recently, Aluthge and

Wang [2]

showed the following results on powers ofp-hyponormaloperators.

THEOREM

A. [2]

^{Let Tbea}p-hyponormaloperator

for

^p

^{(0, 1].}

^The

inequalities

(T

ⁿ

Tn)P/n (T’T)

^p

_ ^(TT*)

^p

^(TnTn*)

^pin

hold

for

^allpositive integern.

COROLLARYA.2

[2] If

^Tisap-hyponormaloperator

for

^p

^(0,

^],then

Tⁿisp/n-hyponormalforanypositive integer n.

By

Corollary A.2,ifTis ahyponormal operator,thenT²belongsto the class of 1/2-hyponormal operators which is smaller than that of paranormal.

As

amoreprecise result than Theorem

A.

1,Furutaand Yanagida

[11]

obtainedthefollowingresult.

THEOREM A.3

[11,

Theorem

l]

Let Tbeap-hyponormaloperator

for

p

(0, ].

Then

(Tn*Tn)

^(p+l)/n

^>_ (T’T)

^{p+I and}

(TT*)

^p+I

^>_ (TnTn*)

^(p+l)/n

hold

for

^allpositive integern.

TheoremA.3assertsthat the first and thirdinequalitiesof Theorem A.1 hold for thelargerexponents (p

+ 1)/n

^thanp/nⁱⁿTheorem A.1.

In fact,Theorem A.3ensuresTheorem

A.

byL6wner-Heinztheorem forp/(p

+ 1) (0, 1)

^andp-hyponormalityofT.

Onthe otherhand, Fujiiand Nakatsu

[6]

showedthefollowingresult.

THEOREM A.4

[6]

^Foreachpositive integer n,

if

^Tisann-hyponormal operator, thenTⁿishyponormal.

Weremark that TheoremA.1, CorollaryA.2 and Theorem A.3 are results onp-hyponormal operatorsfor p

(0,

1],and Theorem A.4is a resultonn-hyponormaloperators forpositiveinteger^n. Inthispaper, moregenerally,weshalldiscusspowers ofp-hyponormaloperators for positive real number p

>

^0.

POWERS OF p-HYPONORMAL OPERATORS 3

2. MAIN

RESULTS

THEOREM Let Tbe ap-hyponormal operator

for

^p

^>

^{O. Then the}

followingassertionshold."

(1)

T^n*Tⁿ

>_ (T* T)n

^and

(TT*)n ^>_

^{T T,* hold}

for

positive integer n such thatn

<

p

+

^1.

(2) (Tn*Tn)

^(p+l)/n

>_ (T’T)

^{p+I and}

(TT*)

^p+I

^>_ (TnTn*)

^(p+l)/nhold

for

positive integer nsuch thatn

>_

p

+

^1.

COROLLARY 2 Let Tbeap-hyponormaloperator

for

^p

^>

^{O. Then the}

followingassertionshold:

(1)

^Tn*Tⁿ

_

^TnTn*holds

^for

positive integer nsuch thatn

<

p.

(2) (T

^n*

Tn)

^p/n

^>_ (TnTn*)

^p/nholdsforpositiveinteger nsuch thatn

>p.

In other words,

if

^{Tis a}p-hyponormal operator

for

^p

^>

^{0, then}

^T"

^is

min{

1,p/n}-hyponorrnalforanypositive integer n.

Incase pE

(0, 1],

Theorem (resp. Corollary

2)

^meansTheoremA.3 (resp. Corollary

A.2).

Corollary 2 also yields Theorem A.4 in case p n. Theorem and Corollary 2can be rewrittenintothe following Theorem

1’

andCorollary

2’,

respectively.We shallprove Theorem 1’

andCorollary 2’.

THEOREM For some positive integer m, let T be a p-hyponormal operator

for

^rn

<

p

<

m. Then thefollowingassertionshold."

(1) Tn*T

ⁿ

>_ (T’T)

^nand

(TT*)

ⁿ

^>_

^TnTn*

holdforn--1,2,...

,m.

(2) (T"* Tn)

^(p+)/"

> (T’T)

^{p+ and}

(TT*)

^p+

^>_ (T"T"*)

^(p+)/" hold

for n--m+

1,m+2,...

COROLLARY 2 Forsome positive integerm, let Tbeap-hyponormal operator

for

^m

<

p

<_

m. Then thefollowingassertionshold."

(1)

^Tn*Tn

>_

TⁿT^n*holds

for

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

(2) (T

^n*

T")

^p/n

^>_ (T

^nT

"*)p/n

^holds

_for

n m,rn

+

^1,...

We

needthefollowingtheorem inordertogiveaproofof Theorem

.

THEOREMB.1

(Furuta

^inequality

[8]) If

^A

^>

^B

^{> O,}

^then

for

^eachr

^{> O,}

(i) ^Br/2APB r/2)

^1/q

_ (Br/2BPBr/2)

^1/q

and

(ii) (Ar/2APAr/2)

^1/q

(W/2BpW/2)1/q

hold

for

^p

^>

^{0 and q}

^>

^with

⁽¹ +

^r)q

^>

^p

+

^r.

We

remark that TheoremB. yieldsL6wner-Heinztheorem whenwe putr 0in(i)^or(ii)stated above.AlternativeproofsofTheoremB. are given in

[4,15]

and alsoanelementaryonepageproofin

[9].

^Itisshownin

[16]

thatthe domain drawn forp, q andrinFig. is the bestpossibleone for Theorem

B.

1.

Proof of

^{Theorem I WeshallproveTheorem byinduction.}

Proof of ⁽¹⁾

^Weshallprove

and

T^n"Tⁿ

(T’T)

ⁿ

(2.1)

(TT*)" ^> T"T

ⁿ

(2.2)

forn 1,2,...,m.

(2.1)

^and

(2.2)

^{alwayshold forn 1.}

Assume

that

(2.1)

and

(2.2)

hold for somen

<

m 1.Then we have

T

n*T

ⁿ

(T’T)

ⁿ

(TT*)

ⁿ

^TnT

^n*

(2.3)

andthe secondinequalityholdsbyp-hyponormality ofTandL6wner- HeinztheoremfornipE

(0, 1]. By (2.3),

^wehave

T^n*Tⁿ

>_ (TT*)" (2.4)

(1,1)

(1 + ^r)q

^p

+

^r

/(,0)

FIGURE

POWERS OFp-HYPONORMAL OPERATORS 5

and

(7-*r)

ⁿ

^>_

(2.4)

^ensures

Tn+l Tn+ T*(T

ⁿ

Tn)T ^>_ T*(TT*)nT (T’T) n+,

and

(2.5)

^ensures

(TT*)

ⁿ⁺

T(T*T)nT ^>_ T(TnTn*)T

^*--

Tn+ Tn+l *.

Hence

(2.1)

^and

(2.2)

^{hold for n}

+ ^<

^{m, so that the proofof}

⁽¹⁾

^is

complete.

Proof of ⁽²⁾

^Weshallprove

(T

^n*

Tn)(p+)/n >_ (T’T)

^p+I

(2.6)

and

(TT*)

^p+I

^>_ (TnTn*)

^(p+l)/n

(2.7)

forn rn

+

^1,rn

+

2,... Let T=

UI ^T[

^{be the}polar decompositionofT where

ITI=(T*T)

^{/2 and} put

A.=ITI

^{zp/" and}

^{B, [T"*[ p/".}

^We

remark that T*

U*IT*[

^{isalso the}polardecomposition of T*.

(a)

^{Casen rn}

+

^1.

(2.1)

^and

(2.2)

^{forn rn ensure}

(r m*rm)p/m (T’T)

^p

(rr*)

^p

(rmrm*)

^p/m

(2.8)

sincethe first and third inequalities hold by

(2.1), (2.2)

and L6wner- Heinz theorem for p/mE

(0, 1],

and the second inequality holds by p-hyponormality ofT.

(2.8)

ensures thefollowing

(2.9)

^and

(2.10).

Am (T

^m*

zm)

^p/m

(ZZ*)

^p

nl. (2.9)

hi (T’T)

^p

(TmTm*)

^p/m Bin.

(2.10)

By

using TheoremB. form/p

>

and 1/p

>

0,wehave

(Tm+

^1.

Tm+l)(p+l)/(m+l) (U*IT*ITm*TmIT*[U)(p+I/(m+I U*([T*ITm*Tm[T*[)(p+/(m+)

U U* II/2pAm/P lctl/2p’(I+I/P)/((m/p)+I/P)U

l+l/p

>_UB U

U*IT*I2(p+I) U

ITI/p+)

(T’T) p+l,

sothat

(2.6)

^{holds forn m}

+

^1.

By

using TheoremB. againform/p

>

and1/p

>

0,wehave

(Tm+ 1Tm+I*)(P+I)/(m+l) (UITITmTm*[T[U*)(P+I)/(m+

⁾

U([T[TmTm*[T[)(P+)/(m+)U*

u(Alll2PBnlPAII2p)(I+llp)I((mlp)+llP)u

< UAI+I/pu

UITI ^/p+/U*

T*I2(P+

(TT*) p+l,

sothat

(2.7)

^{holds forn m}

+

^1.

(b) Assume

that

(2.6)

^and

(2.7)

hold for somen

>

m

+

^1.Then

(2.6)

^and

(2.7)

^fornensure

(T

^n*

Tn)P/n (T’T)

^p

(TT*)

^p

. ^{(TnTn*)P/n (2.11)}

since the firstand third inequalities hold by

(2.6)

^and

(2.7)

^{forn and} L6wner-Heinztheorem forp/(p

+ ¹⁾

^E

^{(0, 1),}

^{and the}second inequality holdsbyp-hyponormality ofT.

(2.11)

^ensuresthe following

(2.12)

^and

(2.13).

An (T

^n*

Tn)

^p/n

^>_ (TT*)

^p

B1. (2.12)

A1 (T’T)

^p

^>_ (TnTn*)

^p/n

_Bn. (2.13)

POWERS OFp-HYPONORMAL OPERATORS 7

By

using Theorem

B.

fornip

>

and1/p

_ ^O,

^wehave

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

¹⁾

(U*IT*ITn*TnlT*[U)(p+)/(n+I) U*(lr*lrn*rnlr*l)(P+l)/(n+l)

^U

U*

(Bll/2pA/pBll/2P)

(I+I/p)/((n/p)+I/p)S

_ U*BI

^+I/p

^U

U*[T*[

^2(p+1)U

IT[-(p+

⁾

(T’T) p+I,

sothat

(2.6)

^{holds forn}

+

^1.

By

using Theorem

B.

again fornip

>

and 1/p

>

0,wehave

(zn+ zn+l*)(p+l)/(n+l) (U[T[TnT*[T[U*)(p+I)/(+

⁾

U(ITITnTn*ITI)(P+I)/(n+I)u

1/2p)(l+l/p)/((n/p)+l/p)

U*

U(AI/2PB/PA1

-<

^{UA +}

^U*

U[T]

^2(p+l)U*

[T*](P+I/

(TT*)p+I,

sothat

(2.7)

^{holds forn}

+

^1.

By (a)

^and

(b), (2.6)

^and

(2.7)

^{hold forn m}

+

^1,m

+

^2,...,that is, the proofof

(2)

iscomplete.

Consequentlytheproofof Theorem 1’iscomplete.

Proof of

^Corollary

Proof of ^{(1) By (1)}

^Theorem

^1’,

^{forn 1,2,...,m 1,}

T

n*T

ⁿ

_{> (T’T)}

ⁿ

>_ (TT*)

ⁿ

^{>_ TnT}

ⁿ

hold since the second inequality holds by p-hyponormality of Tand L6wner-Heinztheorem for nipE

(0, 1).

ThereforeT^n*T

>_

TⁿT^n*holds forn-1,2,...,m- 1.

Proof of ^{(2) By (1)}

^{ofTheorem 1’ and} L6wner-Heinz theorem for p/mE

(0, 1]

ⁱⁿcase n m, andby

(2)

^{of Theorem 1’and}L6wner-Heinz theorem forp/(p

+ 1)

^E

(0, 1)

in casen m

+

^1,m

+

^2,...,

(T

^n"

Tn)P/n (T’T)

^p

(TT*)

^p

(TnTn’)

^pin

hold since the second inequality holds by p-hyponormality ^of T.

Therefore

(T"* T")

^p/n

> (TnT"*)

^p/"holds for n=m,m

+

^1,...

3. BEST

POSSIBILITIES

OF

THEOREM

1 AND

COROLLARY

2 Furuta and Yanagida

[11

^{discussedthe best}possibilities ofTheorem A.3 andCorollaryA.2onp-hyponormaloperators for p

(0, 1]. In

^{this sec-} tion, moregenerally,weshall discuss the best possibilities of Theorem andCorollary2onp-hyponormaloperators for p

>

0.

THEOREM 3 Letnbea positive integersuch thatn

>

2,p

>

^{0 anda}

>

^1.

(1)

^{Incase n}

<

p

+

^{1, thefollowingassertionshold:}

(i) ^Thereexistsap-hyponormaloperatorTsuch that

(’,’ 7"n) (7"* ) ""

(ii)

^Thereexistsap-hyponormaloperatorTsuch that

(VT*)

ⁿ

(TnTn’) .

(2)

^{Incase n} p

+

^{1,thefollowingassertionshold."}

(i) ^{Thereexists a}p-hyponormaloperatorTsuch that

(Tn*Tn)

^((p+l)a)/n

(T’T) ^(p+l)a.

(ii) ^Thereexistsap-hyponormaloperatorTsuch that

(TT*)

^(+l)a

(TnTn*) ((p+l)a)/n.

THEOREM4 Letnbeapositiveintegersuchthatn 2,p

>

^{0 anda}

>

^1.

(1)

^{In case n}

<

p, there exists a p-hyponormal operator T such that

(n* Tn)" (TnT"*)’.

(2)

In case n

p,

there exists a p-hyponormal operator T such that

(Tn*Tn) (T"Tn*) ^p‘/".

Theorem 3(resp.Theorem

4)

assertsthe bestpossibilityof Theorem (resp. Corollary

2).

^We need the following results to give proofs of Theorem 3 andTheorem 4.

POWERSOFp-HYPONORMAL OPERATORS 9

THEOREM C.1

[17,19]

Letp>O,q>O,r>Oand6>O. IfO<q<l ^or

(6 +

^r)q

^<

^p

+

r, then thefollowingassertionshold:

(i) ^Thereexistpositive invertible operators A and

B

on

2

such that A^e

>

B^eand

(Br/2APBr/2)

^1/q

_

^B(p+r)/q.

(ii) There existpositive invertible operatorsA andB on ^]12 such that A^e

>

B^eand

A^(p+r)/q

_ (Ar/2BPAr/2)I/q.

LEMMAC.2

[11]

ForpositiveoperatorsAandB,

define

^{theoperatorTon}

k-o

^Has

follows."

0

B1/2

All2

0

A1/2

0

(3.1)

where[2] showstheplace

of

^the

^{(0, 0)}

^matrixelement. Then thefollowing assertionholds:

(i) T^isp-hyponormal

for

^p

^>

⁰

if

^andonly

if

^Ap

^>_

^B

^p.

Furthermore,thefollowingassertionshold

for 3 ^>

^{0 andintegers n}

^>_

^2:

(ii)

(T * _T")

^/

>_ (T* T) if

^andonly

if

(Bk/2An-kBk/2)

^3/n

^>_

^{B3 holds}

for

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

(iii) (TT*)3 ^>_

^TnTn

)3/,, _if

^andonly

_if

(3.2)

A³

>_ (Ak/2Bn-kAk/2)3/n

^holds

_for

k- 1,2,... ,n- 1.

(3.3)

(iv) (T

ⁿ

Tn)

^/n

^>_ (TnTn*)

^/n

if

^andonly

if

A

>B

holds and

(Bk/2An-kBk/2)/n

^{B and A3}

(Ak/2Bn-kAk/2)/n

hold

for

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

^(3.4)

Proof of

^{Theorem 3 Letn}

^>

^2,p

^>

^0anda

^>

^1.

Proof of ⁽¹⁾

Put pl=n-l>O, ql-1/aE(O,

1), rl--l>O

^and 8=p>O.

Proof of

(i)

By

(i)^{of Theorem}C.1, there existpositiveoperatorsAand BonHsuchthatA⁶

_>

B⁶and

(Brd2AP’Brd2)

^{1/q’ B}

(p’+r’)/q’,

that is,

and

A^p B^p

(3.5)

(B1/2An-1B1/2) _

^Bha.

^(3.6)

Define anoperator Ton

)k-

^Has

(3.1).

^{Then Tis}p-hyponormal by

(3.5)

^and

(i)

^of

Lemma

C.2, and

(T

^n*

Tn) _ ^(T’T)

^{n by (ii) of}

LemmaC.2 sincethecasek of

(3.2)

^doesnothold

for/3

naby

(3.6).

Proofof(ii)

By

(ii)ofTheoremC.1, there exist positive operatorsAand BonHsuchthat

A

^e

>

B^eandA^(pl+rl)/ql

_ (Ar’/2BP’Ar’/2) ^1/q’,

that is, and

A^p

>_

B^p

(3.7)

Aⁿ

(A1/2on-lA1/2)a. (3.8)

Define anoperatorTon

)k-

^Has

(3.1).

^ThenTisp-hyponormalby

(3.7)

^and(i)^of

Lemma

C.2,and

(TT*)

ⁿ

(TnTn*)

by (iii)^ofLemma C.2sincethecasek of

(3.3)

^doesnothold

for/3

^naby

(3.8).

Proof of (2)

Put_Pl n-

> O,

ql n/((p

+ 1)a)> O,

rl

>

^{0 and}

8 =p

> O,

thenwehave

(8 +

Proof of

(i)

By

(i)^{of Theorem}C.1, there existpositiveoperatorsAand BonHsuch thatA^e

>

B^eand

(Br’/2APBr/z)

^{1/ql B}

^(p+r)/q.,

^{that is,}

A^p

>_

B^p

(3.9)

POWERS OFp-HYPONORMALOPERATORS 11

and

(B1/2An-IB1/2)

^{((p+l))/n B(p+l)a.}

(3.10)

Defineanoperator Ton

)k-o

^Has

(3.1).

^ThenTisp-hyponormal by

(3.9)

and(i)ofLemma C.2,and

(T"* Tn)

^((p+l))/n

_ ^(T’T)

^(p+I)by

⁽ⁱⁱ⁾

ofLemmaC.2 sincethecasek

of(3.2)

^doesnothold

for/3

(p

+ 1)a

by

(3.10).

Proof ^{of(ii) By}

^{(ii)ofTheoremC. 1,}there existpositive operators Aand BonHsuch thatA⁶

>

B⁶andA^(p+rl)/ql

(Ar’/2BPArd2) ^1/q’,

that is,

and

A^p

>_

B^p

(3.11)

A^(p+l)a

_ (A1/2Bn-IA1/2)((P+I))/n. (3.12)

Defineanoperator Ton

)k-

^Has

(3.1).

^ThenTisp-hyponormal by

(3.11)

^and(i)^of

Lemma

C.2,and

(TT*)

^(p+I)’

(TnTn*)

^((p+))/nby

(iii)

ofLemmaC.2sincethecasek

of(3.3)

^doesnothold

for/3

(p

+ 1)a

by

(3.12).

Proof of

^{Theorem 4 Letn}

^>

^2,p

^>

^{0 anda}

^>

^1.

Proof of ⁽¹⁾

^Put Pl n

>

^0, ql

1/a

E

(0, 1),

rl 1

>

0 and p

>

0.

By

(i)^{of Theorem}C. 1,there existpositiveoperatorsAandB onHsuch that

A

⁶

>_

B⁶and

(Br’/2APBrd2)

^1/q

_

^B

(p’+r’)/q’,

thatis, and

A^p

>_

B^p

(3.13)

(B1/An-1B1/2)

^B

^n’. (3.14)

Define anoperator Ton

)k-o

^Has

(3.1).

^{Then Tis}p-hyponormal by

(3.13)

^and

(i)

of

Lemma

C.2, and

(T

^n*

Tn) (T"Tn*)

by (iv) of LemmaC.2 since the casek of the second inequality of

(3.4)

^doesnot

hold

for/3

^naby

(3.14).

Proof of ⁽²⁾

^Itiswell known that thereexistpositive operatorsAandB onHsuch that

A^p

>_

B^p

(3.15)

12 M.ITO

and

A^p’

_

^Bp’

^(3.16)

Define anoperator Ton

)k=_

^Has

^(3.1).

^{Then Tis}p-hyponormal by

(3.15)

^and

(i)

of Lemma C.2,and

(T

^n*

Tn)

^p/n

(TnTn*)

^p/nby (iv) of LemmaC.2sincethe first inequality of

(3.4)

^doesnothold

for/3

pc by

(3.16).

4.

CONCLUDING

REMARK

An

operator T is said to be log-hyponormal if T is invertible and log T*T>_logTT*. It is easily obtained that every invertible p- hyponormal operator is log-hyponormal ^since log is an operator monotonefunction.Weremark thatlog-hyponormalcanbe regarded as0-hyponormalsince

(T* T)

^p

>_ _(TT*)

^papproaches logT*T> logTT*

as p^--,+0.

As

an extension of Theorem A.1, Yamazaki

[18]

obtained the following Theorem D.1 and Corollary D.2 on log-hyponormal operators.

THEOREM

D. [18]

^{Let Tbe a} log-hyponormal operator. Then the following inequalities hold

for

^allpositive integer^n."

(1)

^T*T

_ ^(T2*T2)

^1/2

^_...

^(_

^{(Tn’Tn) 1In.}

(2)

^TT*

_ (TZT2")

^/2

>_... >_ (rnrn’) ^/n.

COROLLARYD.2

[18] If

^Tisalog-hyponormaloperator,then T isalso log-hyponormal

for

^anypositive integer n.

ThebestpossibilitiesofTheoremD. andCorollaryD.2arediscussed in

[12].

As

aparallelresulttoTheoremD.1,FurutaandYanagida

[12]

^showed thefollowingTheorem D.3onp-hyponormaloperators for pE

(0, ].

THEOREMD.3

[12]

Let

Tbeap-hyponormaloperatorforp (0, 1].

^Then thefollowinginequalities hold

for

^allpositive integer^n."

(1) (T’T)

^p+

^<_ (T

^2’

T2)

^(p+l)/2

^<_... <_ (T

^n*

Tn) (p+l)/n.

(2) (TT*)

^p+I

^>_ (TT’)

^(p+l)/z

^{>_ >_} (TnTn*) ^(p+l)/n.

POWERSOFp-HYPONORMALOPERATORS 13

In fact,Theorem D.3 in thecasep /0corresponds^toTheoremD.1.

As

a further extension of Theorem D.3, we obtain the following Theorem 5onp-hyponormaloperators for p

>

0.

THEOREM 5 For some positive integer m, let T be ap-hyponormal operator

for

^rn-

^<

^p

^<

^m. Then the following inequalities hold

for

n=m/1,m+2,..."

(1) (T*T)p+I<_ _(T

^m+l*

_Tm+l)

(p+l)/(m+l)

<_ (T

^m+2*

Tin+2)

(p+l)/(m+2)

<_’"

(Tn*Tn)(p+I)/n.

(2) (TT*)P+I> (T

^m+l

Tm+l*)(P+l)/(m+l) ^>_ (Tm+2Tm+2*)(P+I)/(m+2)

^>_"

^>_

(TnTn*)(P+I)/n.

We

remark that Theorem 5yieldsTheorem D.3by puttingrn 1.

Scrutinizing theproofofTheoremD.1 and TheoremD.3,werecognize that the following resultplaysanimportant role.

THEOREM D.4

[12,18]

LetTbeap-hyponormaloperatorforpE

(0, 1]

^or

a log-hyponormaloperator. Then the following inequalities hold

for

^all

positive integer n:

(1) [Tn+l[2n/(n+l) ^>_ IT"l,i.e., (Tn+l*Tn+l)n/(n+l) ^{>_ Tn*T} n.

(2) [Tn*[

²

^_> [Tn+l*[ 2n/(n+l),

i.e., TnT^n*

>_ (Tn+lzn+l*)n/(n+l).

Weremark that itwasshown in

[14]

that TheoremD. and Theorem D.4holdevenifaninvertible operatorT belongstoclassA (i.e.,

TZl ^>_

TI ⁾

^{which was introduced in}

^[10]

^{as a class} of operators including p-hyponormalandlog-hyponormaloperators.

Proof of

^{Theorem 5}

^It

^{is easilyobtainedby}L6wner-Heinztheorem that Theorem D.4remainsvalid forp-hyponormal operatorsfor p

>

^0.

Proof of (1) By (1)

of Theorem D.4 and L6wner-Heinz theorem for (p

+ ^1)In

^E

^{(0, 1),}

(T

^n+l*

zn+l)

(p+l)/(n+l)

(T

^n*

Tn)

^(p+l)/n

(4.1)

holds forn m

+

^1,m

+

^{2,... Then}

(T’T)

^p+I

< (T

^m+l*

Tm+l)(p+l)/(m+

¹⁾

< ^(T

^m+2*

Tm+2)(p+l)/(m+

²⁾

< ^{<_ (T}

^n*

^Tn)(p+l)/n

holdsby

(2)

^{of Theorem and}

(4.1).

Proof of ^{(2) By (2)}

^{of Theorem D.4}and L6wner-Heinz theorem for (p

+ ^1)In

^E

^{(0, 1),}

(TnTn*)

^(p+l)/n

^>_ (Tn+l Tn+l*)

(p+l)/(n+l)

(4.2)

holdsforn rn

+

^1,rn

+

^{2,... Then}

(TT*)

^p+]

^> (T

^m+l

Tm+l*)(p+l)/(m+l) _ (Tm+2Tm+2*)(p+l)/(m+2)

>...

> (TnTn*)

^(p+l)/n

holdsby

(2)

^{of Theorem1’and}

(4.2).

Acknowledgement

The author would like to express his cordial thanks to Professor Takayuki

Furuta

for his kindly guidance and encouragement.

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