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volume 7, issue 1, article 32, 2006.

Received 16 May, 2005;

accepted 26 November, 2005.

Communicated by:S.S. Dragomir

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Journal of Inequalities in Pure and Applied Mathematics

POWERS OF CLASS wF(p, r, q)OPERATORS

JIANGTAO YUAN AND CHANGSEN YANG

LMIB and Department of Mathematics Beihang University

Beijing 100083, China

EMail:yuanjiangtao02@yahoo.com.cn

College of Mathematics and Information Science Henan Normal University

Xinxiang 453007, China.

EMail:yangchangsen117@yahoo.com.cn

c

2000Victoria University ISSN (electronic): 1443-5756 152-05

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Powers of ClasswF(p, r, q) Operators

Jiangtao Yuan and Changsen Yang

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J. Ineq. Pure and Appl. Math. 7(1) Art. 32, 2006

http://jipam.vu.edu.au

Abstract

This paper is to discuss powers of classwF(p, r, q)operators for1≥ p > 0, 1≥r >0andq≥1; and an example is given on powers of classwF(p, r, q) operators.

2000 Mathematics Subject Classification:47B20, 47A63.

Key words: ClasswF(p, r, q), Furuta inequality.

Supported in part by NSF of China(10271011) and Education Foundation of Henan Province(2003110006).

1. Introduction

LetHbe a complex Hilbert space andB(H)be the algebra of all bounded linear operators inH, and a capital letter (such asT) denote an element ofB(H). An operatorT is said to bek-hyponormal fork >0if(T^∗T)^k ≥(T T^∗)^k, whereT^∗ is the adjoint operator of T. Ak-hyponormal operatorT is called hyponormal if k = 1; semi-hyponormal if k = 1/2. Hyponormal and semi-hyponormal operators have been studied by many authors, such as [1,11, 16, 20, 21]. It is clear that every k-hyponormal operator isq-hyponormal for0 < q ≤ kby the Löwner-Heinz theorem (A≥B ≥0ensuresA^α ≥B^αfor any1≥α≥0). An invertible operator T is said to belog-hyponormal iflogT^∗T ≥ logT T^∗, see [18, 19]. Every invertiblek-hyponormal operator fork > 0islog-hyponormal since logt is an operator monotone function. log-hyponormality is sometimes regarded as 0-hyponormal since(X^k−1)/k →logX ask →0forX >0.

As generalizations of k-hyponormal and log-hyponormal operators, many authors introduced many classes of operators, see the following.

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Definition A ([5,6]).

(1) Forp > 0andr >0, an operatorT belongs to classA(p, r)if (|T^∗|^r|T|^2p|T^∗|^r)^p+r^r ≥ |T^∗|^2r.

(2) Forp > 0, r ≥0andq ≥1, an operatorT belongs to classF(p, r, q)if (|T^∗|^r|T|^2p|T^∗|^r)¹^q ≥ |T^∗|^2(p+r)^q .

For eachp >0andr >0, classA(p, r)contains allp-hyponormal and log- hyponormal operators. An operatorT is a classA(k)operator ([9]) if and only if T is a class A(k,1) operator,T is a class A(1) operator if and only ifT is a class Aoperator ([9]), and T is a class A(p, r)operator if and only ifT is a classF p, r,^p+r_r

operator.

Aluthge-Wang [3] introduced w-hyponormal operators defined by

∼

T ≥

|T| ≥

∼

T

∗

where the polar decomposition ofT isT =U|T|and

∼

T =|T|^1/2U|T|^1/2 is called the Aluthge transformation ofT. As a generalization ofw-hyponormality, Ito [12] and Yang-Yuan [25,26] introduced the classeswA(p, r)andwF(p, r, q) respectively.

Definition B.

(1) Forp > 0, r >0,an operatorT belongs to classwA(p, r)if

(|T^∗|^r|T|^2p|T^∗|^r)^p+r^r ≥ |T^∗|^2r and |T|^2p ≥(|T|^p|T^∗|^2r|T|^p)^p+r^p .

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(2) Forp > 0, r ≥0, andq ≥1, an operatorT belongs to classwF(p, r, q)if (|T^∗|^r|T|^2p|T^∗|^r)¹^q ≥ |T^∗|^2(p+r)^q and |T|^2(p+r)(1−¹^q⁾≥(|T|^p|T^∗|^2r|T|^p)¹⁻¹^q, denoting(1−q⁻¹)⁻¹ byq^∗ (whenq >1) becauseqand(1−q⁻¹)⁻¹ are a couple of conjugate exponents.

An operatorT is aw-hyponormal operator if and only ifT is a classwA(¹₂,¹₂) operator,T is a classwA(p, r)operator if and only ifT is a classwF(p, r,^p+r_r ) operator.

Ito [15] showed that the classA(p, r)coincides with the classwA(p, r)for eachp > 0andr > 0, class Acoincides with classwA(1,1). For eachp > 0, r ≥ 0 and q ≥ 1 such that rq ≤ p+r, [25] showed that class wF(p, r, q) coincides with classF(p, r, q).

Halmos ([11, Problem 209]) gave an example of a hyponormal operatorT whose square T² is not hyponormal. This problem has been studied by many authors, see [2,10,14,22,27]. Aluthge-Wang [2] showed that the operatorTⁿ is(k/n)-hyponormal for any positive integernifT isk-hyponormal.

In this paper, we firstly discuss powers of class wF(p, r, q) operators for 1 ≥ p > 0, 1 ≥ r > 0and q ≥ 1. Secondly, we shall give an example on powers of classwF(p, r, q)operators.

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2. Result and Proof

The following assertions are well-known.

Theorem A ([15]). Let 1 ≥ p > 0, 1 ≥ r > 0. ThenTⁿ is a classwA(_n^p,_n^r) operator.

Theorem B ([13]). Let1≥p >0,1≥r≥0,q≥1andrq ≤p+r. IfT is an invertible classF(p, r, q)operator, thenTⁿis aF(_n^p,^r_n, q)operator.

Theorem C ([25]). Let 1 ≥ p > 0, 1 ≥ r ≥ 0; q ≥ 1 when r = 0 and

p+r

r ≥q ≥1whenr >0. IfT is a classwF(p, r, q)operator, thenTⁿis a class wF(_n^p,_n^r, q)operator.

Here we generalize them to the following.

Theorem 2.1. Let1 ≥p > 0,1 ≥ r >0;q > ^p+r_r . IfT is a classwF(p, r, q) operator such thatN(T)⊂N(T^∗), thenTⁿis a classwF(_n^p,_n^r, q)operator.

In order to prove the theorem, we require the following assertions.

Lemma A ([8]). Letα∈RandXbe invertible. Then(X^∗X)^α=X^∗(XX^∗)^α−1X holds, especially in the caseα≥1, LemmaAholds without invertibility ofX.

Theorem D ([15]). LetA, B ≥ 0.Then for eachp, r≥ 0, the following asser- tions hold:

(1) B^r²A^pB^r²_p+r^r

≥B^r ⇒ A^p²B^rA^p²_p+r^p

≤A^p. (2) A^p²B^rA^p²_p+r^p

≤A^pandN(A)⊂N(B)⇒ B^r²A^pB^r²_p+r^r

≥B^r.

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Theorem E ([24]). LetT be a classwAoperator. Then|Tⁿ|ⁿ² ≥ · · · ≥ |T²| ≥

|T|² and|T^∗|² ≥ |(T²)^∗| ≥ · · · ≥ |(Tⁿ)^∗|ⁿ² hold.

Theorem F ([25]). LetT be a classwF(p₀, r₀, q₀)operator forp₀ >0, r₀ ≥0 andq₀ ≥1. Then the following assertions hold.

(1) Ifq ≥q₀ andr₀q ≤p₀ +r₀, thenT is a classwF(p₀, r₀, q)operator.

(2) If q^∗ ≥ q₀^∗, p₀q^∗ ≤ p₀ + r₀ and N(T) ⊂ N(T^∗), then T is a class wF(p₀, r₀, q)operator.

(3) Ifrq ≤p+r, then classwF(p, r, q)coincides with classF(p, r, q).

Theorem G ([25]). Let T be a class wF

p₀, r₀,^p_δ⁰^+r⁰

0+r0

operator for p₀ > 0, r₀ ≥ 0 and −r₀ < δ₀ ≤ p₀. Then T is a class wF

p, r,_δ^p+r

0+r

operator for p≥p0 andr≥r0.

Proposition A ([25]). Let A, B ≥ 0; 1 ≥ p > 0, 1 ≥ r > 0; ^p+r_r ≥ q ≥ 1.

Then the following assertions hold.

(1) If B^r²A^pB^r²¹_q

≥B^p+r^q andB ≥C, then C^r²A^pC^r²¹_q

≥C^p+r^q . (2) IfB^p+r^q ≥ B^r²C^pB^r²¹_q

,A≥Band the condition (*) if lim

n→∞B¹²x_n = 0 and lim

n→∞A¹²x_nexists, then lim

n→∞A¹²x_n= 0 holds for any sequence of vectors{x_n}, thenA^p+r^q ≥ A^r²C^pA^r²¹_q

.

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Proof of Theorem2.1. Put δ = ^p+r_q −r, then−r < δ < 0by the hypothesis.

Moreover, if

(|T^∗|^r|T|^2p|T^∗|^r)^p+r^r+δ ≥ |T^∗|^2(r+δ) and |T|^2(p−δ) ≥(|T|^p|T^∗|^2r|T|^p)^p−δ^p+r, then T is a classwA operator by Theorem Gand Theorem D, so that the following hold by takingA_n=|Tⁿ|²ⁿ andB_n =|(Tⁿ)^∗|ⁿ² in TheoremE

(2.1) A_n≥ · · · ≥A₂ ≥A₁ and B₁ ≥B₂ ≥ · · · ≥B_n.

Meanwhile,A_nandA₁ satisfy the following for any sequence of vectors{x_m} (see [24])

if lim

m→∞A

1 2

1x_m = 0 and lim

m→∞A

1

n2x_mexists, then lim

m→∞A

1

n2x_m = 0.

Then the following holds by PropositionA (A_n)^p+r^q^∗ ≥

(A_n)^p²(B₁)^r(A_n)^p²_q¹∗

≥

(A_n)^p²(B_n)^r(A_n)^p²_q¹∗

,

and it follows that

|Tⁿ|^2(p+r)^nq^∗ ≥

|Tⁿ|ⁿ^p|(Tⁿ)^∗|^2rⁿ|Tⁿ|ⁿ^p_q¹∗

.

We assert thatN(T)⊂N(T^∗)impliesN(Tⁿ)⊂N((Tⁿ)^∗).

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In fact,

x∈N(Tⁿ) ⇒Tⁿ⁻¹x∈N(T)⊆N(T^∗)

⇒Tⁿ⁻²x∈N(T^∗T) = N(T)⊆N(T^∗)

· · ·

⇒x∈N(T)⊆N(T^∗)

⇒x∈N(T^∗)⊆N((Tⁿ)^∗), thus

|(Tⁿ)^∗|ⁿ^r|Tⁿ|^2pⁿ|(Tⁿ)^∗|ⁿ^r¹_q

≥ |(Tⁿ)^∗|^2(p+r)^nq

holds by Theorem D and the Löwner-Heinz theorem, so that Tⁿ is a class wF(_n^p,_n^r, q)operator.

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3. An Example

In this section we give an example on powers of classwF(p, r, q)operators.

Theorem 3.1. LetAandB be positive operators onH,U andDbe operators onL∞

k=−∞H_k,whereH_k∼=H,as follows

U =





 . ..

. .. 0 1 0

1 (0) 1 0

1 0 . .. ...





 ,

D=





 . ..

B¹² B¹²

(A¹²) A¹²

A¹² . ..





 ,

where(·)shows the place of the(0,0)matrix element, andT =U D. Then the following assertions hold.

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(1) If T is a classwF(p, r, q)operator for1 ≥p > 0,1 ≥r ≥ 0,q ≥ 1and rq ≤p+r, thenTⁿis awF(^p_n,_n^r, q)operator.

(2) IfT is a classwF(p, r, q)operator such thatN(T)⊂N(T^∗),1≥p >0, 1≥r ≥0,q≥1andrq > p+r, thenTⁿis awF(_n^p,_n^r, q)operator.

Remark 1. Noting that Theorem3.1holds without the invertibility ofAandB, this example is a modification of ([4], Theorem 2) and ([23], Lemma 1).

We need the following well-known result to give the proof.

Theorem H (Furuta inequality [7], in brief FI). IfA≥B ≥0, then for each r ≥0,

(B^r²A^pB^r²)¹^q ≥(B^r²B^pB^r²)¹^q (i)

and

(A^r²A^pA^r²)¹^q ≥(A^r²B^pA^r²)¹^q (ii)

hold forp≥0andq≥1with(1 +r)q≥p+r.

TheoremHyields the Löwner-Heinz inequality by puttingr= 0in (i) or (ii) of FI. It was shown by Tanahashi [17] that the domain drawn forp, q andrin the Figure is the best possible for TheoremH.

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p

(1,0) q (0,−r)

(1,1)

q = 1 p =q

(1 + r)q = p + r

Proof of Theorem3.1. By simple calculations, we have

|T|² =





 . ..

B B

(A) A

A . ..





 ,

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|T^∗|² =





 . ..

B B

(B) A

A . ..





 ,

therefore

|T^∗|^r|T|^2p|T^∗|^r =





 . ..

B^p+r

(B^r²A^pB^r²) A^p+r

A^p+r . ..







and

|T|^p|T^∗|^2r|T|^p =





 . ..

B^p+r

(A^p²B^rA^p²) A^p+r

A^p+r . ..





 ,

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thus the following hold forn≥2

Tⁿ^∗Tⁿ

=





 . ..

Bⁿ Bⁿ

Bⁿ⁻¹² ABⁿ⁻¹² . ..

B^j²A^n−jB^j² . ..

B¹²Aⁿ⁻¹B¹² (Aⁿ)

Aⁿ . ..







and

TⁿTⁿ^∗

=





 . ..

Bⁿ (Bⁿ)

A¹²Bⁿ⁻¹A¹² . ..

A^j²B^n−jA^j² . ..

Aⁿ⁻¹² BAⁿ⁻¹² Aⁿ

Aⁿ . ..







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Proof of (1).T is a classwF(p, r, q)operator is equivalent to the following (B^r²A^pB^r²)¹^q ≥B^p+r^q and A^p+r^q^∗ ≥(A^p²B^rA^p²)^q¹^∗,

Tⁿbelongs to classwF(^p_n,_n^r, q)is equivalent to the following (3.1) and (3.2).

(3.1)











(B^r²(B²^jA^n−jB^j²)ⁿ^pB^r²)¹^q ≥B^p+r^q (B^r²A^pB^r²)¹^q ≥B^p+r^q

A²^jB^n−jA^j²_2n^r A^p

A^j²B^n−jA^j²_2n^r ¹_q

≥

A^j²B^n−jA^j²^p+r_nq where j = 1,2, ..., n−1.

(3.2)











B^j²A^n−jB^j² _2n^p

B^r

B^j²A^n−jB^j² _q¹∗

≥

B^j²A^n−jB^j² ^p+r_nq∗

A^p+r^q^∗ ≥(A^p²B^rA^p²)^q¹^∗

A^p+r^q^∗ ≥

A^p²

A^j²B^n−jA^j²^r_n A^p²

_q¹∗

where j = 1,2, ..., n−1.

We only prove (3.1) because of TheoremD.

Step 1. To show

B^r²

B^j²A^n−jB^j²^p_n B^r²

¹_q

≥B^p+r^q

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forj = 1,2, ..., n−1.

In fact,T is a classwF(p, r, q)operator for1≥p >0,1≥r ≥0,q≥1and rq ≤ p+rimpliesT belongs to classwF

j, n−j,_δ+jⁿ

,whereδ = ^p+r_q −r by TheoremGand TheoremD, thus

B^j²A^n−jB²^j^δ+j_n

≥B^δ+j and A^n−j−δ ≥

A^n−j² B^jA^n−j² ^n−j−δ_n

Therefore the assertion holds by applying (i) of TheoremHto

B^j²A^n−jB²^j^δ+j_n andB^δ+j for

1 + _δ+j^r

q≥ _δ+j^p + _δ+j^r . Step 2. To show

A^j²B^n−jA^j²_2n^r A^p

A^j²B^n−jA²^j_2n^r ¹_q

≥

A²^jB^n−jA^j²^p+r_nq

forj = 1,2, ..., n−1.

In fact, similar to Step 1, the following hold

B^n−j² A^jB^n−j² ^δ+n−j_n

≥B^δ+n−j and A^j−δ≥

A^j²B^n−jA²^j^j−δ_n ,

this implies that A^j ≥

A^j²B^n−jA^j²^j_n

by Theorem D. Therefore the assertion holds by applying (i) of Theorem H to A^j and

A^j²B^n−jA^j² _n^j

for (1 + ^r_j)q ≥ ^p_j + ^r_j.

Proof of (2). This part is similar to Proof of (1), so we omit it here.

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We are indebted to Professor K. Tanahashi for a fruitful correspondence and the referee for his valuable advice and suggestions, especially for the improve- ment of Theorem2.1.

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