ofp-Hyponormal Powers

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Powers ofp-Hyponormal Operators

ARIYADASA ALUTHGE^aand DERMINGWANG

aDepartmentofMathematics,Marshall University, Huntington, VVV25-755-2560,USA,"^bDepartmentof Mathematics, CaliforniaStateUniversity,LongBeach,CA90840-1001,USA (Received20 November 1997; Revised 9 July1998)

ApplyingFuruta’s andHansen’sinequalities, it is shownthatifTis ap-hyponormal operator, thenT is(p/n)-hyponormal. Applicationsare obtained.

Keywords: p-Hyponormal operator;Furuta’sinequality;Hansen’sinequality 1991AMS SubjectClassifications: ^Primary47B20, 47A30

1

INTRODUCTION

Let H be a complex Hilbert space and

L(H)

^{be the} algebra of bounded linear operators on H.

An

operator TE

L(H)

^{is said to} be p-hyponormal,p

>

^0,if

(T’T)

^p

^>_ (TT*) ^{p. A}

p-hyponormaloperatoris said to be hyponormal if p 1; semi-hyponormal if p

1/2.

^{The well}

known L6wner-Heinz inequality implies that every p-hyponormal operator is q-hyponormal for any 0

<

q

<p.

Hyponormal operators have beenstudiedbymanyauthors,such as Halmos

[7],

Stampfli

[10,11]

andXia

[13].

Semi-hyponormalitywasintroducedbyXia

[12].

^See

[13]

for properties of semi-hyponormal operators. For p-hyponormal operators,see

[1,2].

Throughoutthis paperweassume0

<

p

<

and use acapital letter to denote an operator in

L(H).

^In

[7,

Problem 164], Halmos gave an example of a hyponormal operator A whose square A² is not

*Correspondingauthor.

279

hyponormal.

Here

weuse

Furuta’s [5]

and

Hansen’s [6]

inequalitiesto show that ifTisp-hyponormal, then T²is(p/2)-hyponormal.

In

fact, wewillshow that for any positive integer n, the operator Tⁿ is (p/n)- hyponormal. Applicationsofourresultarealso obtained.

2

THE RESULT

LEMMA (Furuta’s

inequality

[5]) If

^A

^>_

^B

^>_

^0, then the inequalities

(BrAPBr)

^{1/q B(p+2r)/q}

and

A(p+2r)/q

>_ (ArBPAr)

^1/q

hold

for

^p,r

^>_

^0,q

^_>

^with

(1 +

^2r)q

^>_

^p

+

^2r.

LEMMA

2

(Hansen’s

inequality

[6]) IfA ^>_

^0and

[[B][ ^_<

^{1, then}

(B*AB)

^p

>_ B*APB

forO<_p< 1.

THEOREM Let Tbeap-hyponormaloperator. Theinequalities

(T

^n*

Tn)P/n ^>_ (T’T)

^p

^>_ (TT*)

^p

^>_ (TnTn*)

^p/n

hold

for

^allpositive integern.

Proof

^{Let T=}

UITI

^{be the polar} decomposition of T. For each positive integer n, let

An=(Tn*Tn)

^{p/n and}

Bn=(TnTn*)

^{p/n. We} will useinductiontoestablishthe inequalities

An ^>_ A1 ^>_ B1 ^>_ Bn. (1)

The inequalities

(1)

clearlyhold forn 1.

Assume (1)

^{holdforn k.}

The induction hypothesis and the assumption that Tisp-hyponormal imply

U*AkU ^>_ U’A1

^U

>_ A1.

Let

Ck u*Akk/P ^U) p/k. Hansen’s

inequalityimplies

Ck >_ U*AkU ^>_

A

.

^Thus

Ak+l ^(T

^*+

Tk+l)

^(p/k+l)

T*

T^*kT

k) T)

^(p/k+l)

([TIU*Akk/PUIT[)

^(p/k+l)

(A1/2prk/p

""k

All/2p)(p/k+I)

>_A

by

Furuta’s

inequality.

On

the other hand, the induction hypothesis implies

Bk

^{<_B1 <_A1.}

Thus

where the inequality follows from

Furuta’s

inequality.Therefore,

Ak+l _ A1 >_ B1 >_ Bk+l

and hence, by induction, inequalities

(1)

^{hold forn}

>

1. The proof^is complete.

COROLLARY

If

the operator Tis p-hyponormal, then

T"

is (p/n)- hyponormal.

Concrete examplesofnon-hyponormal p-hyponormaloperatorsare hard to come by.

In [12],

Xia gave an example ofa singularintegral operator which is semi-hyponormal but nothyponormal. Corollary allowsustogive anotherexampleofasemi-hyponormal operator which

is nothyponormal. Let Abe the operatorⁱⁿHalmos’[7, Problem

164].

Thus,

A

ishyponormalbutA²is nothyponormal.

By

Corollary 1,A²is semi-hyponormal.

Moreover,

Aⁿis(1/2n)-hyponormal.

3

APPLICATIONS

In [10, Theorem

5],

StampfliprovedthatifTishyponormaland Tⁿis normal forsomepositive integern,then Tisnormal. Stampfli’s result had been extended by Ando

[3]

to the case where Tis paranormal.

Althoughnot as broad asAndo’s extension, Theorem caneasily be usedtoextend Stampfli’s resulttop-hyponormaloperatorsasfollows.

COIOILARY 2 Let the operator Tbep-hyponormal.

If

^{T isnormal,}

then Tisnormal.

Proof ^By

^{Theorem and}the assumption that

T"

isnormal,

Whence

T’T-

TT*.Theproofiscomplete.

In

[9,

Theorem

7],

Putnam provedthatifTishyponormal,andr

>

0 is such that r

Er(T*T),

then there is a z

Ea(T)

such that

Izl

^=r.

Recently, Ch6 and Itoh

[4,

Theorem

4]

generalized

Putnam’s

result to thecase where the operator Tisp-hyponormal. Theorem can be utilizedtogiveageneralization of the result of Ch6 and Itohasfollows.

THEOIEM 2. Let Tbe ap-hyponormal operator and n be apositive integer.

If

^r

^>_

^{0 is such that r2}

^cr(T

^n*

^Tn),

^{then there is a z}

^or(T)

such that

Izl"-r.

Proof

^{Theorem implies T is} (p/n)-hyponormal. Therefore, by

[4,

Theorem

4],

there is a w

or(T")

such that

Iwl

^{=r. Since}

{z":

z

a(T)},

^{thereis a z E}

or(T)

^suchthatz

"-

w. Clearly

Izl"-

^rand

theproofiscomplete.

As

an extension of the well-known Putnam’s area inequality for hyponormal operators [8], ^Xia [13, Theorem

XI.5.1]

proved the following Theorem 3 for the case in which Tis p-hyponormal with p

_> 1/2

^{andn 1.}In [4,Theorem

5],

Ch6 andItoh extendedXia’sresult

top-hyponormaloperators with 0

<

p

< 1/2.

THEOrtEM 3 Let T be p-hyponormal.

If ^or(T)

^C_

{rei:

0

<

0

<

2Trim}

for

some positive integerm, then

ii(Tn, zn)p/n (TnTn.)p/nll <_ nPTr JJ

_(r)

^p2p-1

^dpdO

for

positive integers n

<

m.

Proof ^By

^{Theorem 1, Tn is} (p/n)-hyponormal. It follows from [4,Theorem

5]

that

I](Tn*Zn)p/n (ZnZ"*)P/nll <- PnTf JY"

r2(p/n)-I drdqS.

o’(Z

Since

cr(T {pneinO:

pe^iOE

or(T)),

the result follows by the substitu- tionsr

pn

and

b

^nO.

Acknowledgment

Thispaperwaswritten whilethefirstauthor, onsabbaticalleave from Marshall University,was avisiting adjunctprofessor^atthe

Department

of Mathematics, ^CaliforniaState University,

Long

Beach, Fall, 1997.

Hewishes toexpresshisdeep gratitudetohis host forwarm hospitality and support.

References

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Math.Soc.,123(1995),^2435-2440.

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+

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