and an example is given on powers of classwF(p, r, q)operators

http://jipam.vu.edu.au/

Volume 7, Issue 1, Article 32, 2006

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

JIANGTAO YUAN AND CHANGSEN YANG LMIBANDDEPARTMENT OFMATHEMATICS

BEIHANGUNIVERSITY

BEIJING100083, CHINA

[email protected]

COLLEGE OFMATHEMATICS ANDINFORMATIONSCIENCE

HENANNORMALUNIVERSITY

XINXIANG453007, CHINA

[email protected]

Received 16 May, 2005; accepted 26 November, 2005 Communicated by S.S. Dragomir

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

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

2000 Mathematics Subject Classification. 47B20, 47A63.

1. INTRODUCTION

LetH be a complex Hilbert space andB(H)be the algebra of all bounded linear operators in H, and a capital letter (such as T) denote an element of B(H). An operatorT is said to be k-hyponormal for k > 0 if (T^∗T)^k ≥ (T T^∗)^k, where T^∗ is the adjoint operator of T. A k-hyponormal operator T 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 everyk-hyponormal operator isq-hyponormal for0< q ≤kby the Löwner-Heinz theorem (A ≥B ≥ 0ensuresA^α ≥ B^α for any 1≥ α ≥ 0). An invertible operatorT is said to belog-hyponormal iflogT^∗T ≥logT T^∗, see [18, 19]. Every invertiblek- hyponormal operator fork >0islog-hyponormal sincelogtis an operator monotone function.

log-hyponormality is sometimes regarded as 0-hyponormal since(X^k−1)/k→logXask →0 forX >0.

As generalizations ofk-hyponormal andlog-hyponormal operators, many authors introduced many classes of operators, see the following.

ISSN (electronic): 1443-5756

c 2006 Victoria University. All rights reserved.

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

152-05

Definition A ([5, 6]).

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

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

For each p > 0 and r > 0, class A(p, r) contains all p-hyponormal and log-hyponormal operators. An operator T is a class A(k) operator ([9]) if and only if T is a class A(k,1) operator,T is a classA(1)operator if and only ifT is a classAoperator ([9]), andT is a class A(p, r)operator if and only ifT is a classF p, r,^p+r_r

operator.

Aluthge-Wang [3] introducedw-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 trans- formation of T. As a generalization of w-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+rr ≥ |T^∗|^2r and |T|^2p ≥(|T|^p|T^∗|^2r|T|^p)^p+rp . (2) Forp >0, r≥0, andq ≥1, an operatorT belongs to classwF(p, r, q)if

(|T^∗|^r|T|^2p|T^∗|^r)^1q ≥ |T^∗|^2(p+r)q and |T|^{2(p+r)(1−1q)} ≥(|T|^p|T^∗|^2r|T|^p)^1−1q,

denoting (1−q⁻¹)⁻¹ by q^∗ (whenq > 1) because q and(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 > 0and r > 0, class A coincides with class wA(1,1). For each p > 0, r ≥ 0 and q ≥ 1 such that rq ≤p+r, [25] showed that classwF(p, r, q)coincides with classF(p, r, q).

Halmos ([11, Problem 209]) gave an example of a hyponormal operatorT whose squareT² is not hyponormal. This problem has been studied by many authors, see [2, 10, 14, 22, 27].

Aluthge-Wang [2] showed that the operatorTⁿis(k/n)-hyponormal for any positive integern ifT isk-hyponormal.

In this paper, we firstly discuss powers of classwF(p, r, q)operators for1≥p >0,1≥r >

0andq≥1. Secondly, we shall give an example on powers of classwF(p, r, q)operators.

2. RESULT AND PROOF

The following assertions are well-known.

Theorem A ([15]). Let1≥p > 0,1≥r >0. ThenTⁿis a classwA(_n^p,_n^r)operator.

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

Theorem C ([25]). Let1 ≥ p > 0,1 ≥ r ≥ 0; q ≥ 1when r = 0 and ^p+r_r ≥ q ≥ 1when r >0. IfT is a classwF(p, r, q)operator, thenTⁿis a classwF(_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(^p_n,_n^r, q)operator.

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

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

Theorem D ([15]). LetA, B ≥0.Then for eachp, r≥0, the following assertions 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^p andN(A)⊂N(B)⇒ B^r2A^pB^r2_p+r^r

≥B^r.

Theorem E ([24]). Let T be a class wA operator. Then |Tⁿ|ⁿ² ≥ · · · ≥ |T²| ≥ |T|² and

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

Theorem F ([25]). Let T be a classwF(p₀, r₀, q₀)operator for p₀ > 0, r₀ ≥ 0 andq₀ ≥ 1.

Then the following assertions hold.

(1) Ifq≥q₀andr₀q≤p₀+r₀, thenT is a classwF(p₀, r₀, q)operator.

(2) Ifq^∗ ≥q₀^∗,p₀q^∗ ≤p₀+r₀andN(T)⊂N(T^∗), thenT is a classwF(p₀, r₀, q)operator.

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

Theorem G ([25]). LetT be a classwF

p₀, r₀,^p_δ^0+r0

0+r0

operator forp₀ >0,r₀ ≥0and−r₀ <

δ₀ ≤p₀. ThenT is a classwF

p, r,_δ^p+r

0+r

operator forp≥p₀andr≥r₀.

Proposition A ([25]). LetA, 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+rq andB ≥C, then C^r2A^pC^r21_q

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

,A≥B and 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{xn}, thenA^p+rq ≥ A^r2C^pA^r21_q

.

Proof of Theorem 2.1. Putδ = ^p+r_q −r, then−r < δ <0by the hypothesis. Moreover, if (|T^∗|^r|T|^2p|T^∗|^r)^r+δp+r ≥ |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 G and Theorem D, so that the following hold by takingAn =|Tⁿ|ⁿ² andBn =|(Tⁿ)^∗|ⁿ² in Theorem E

(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 Proposition A (An)^p+rq∗ ≥

(An)^p2(B1)^r(An)^p2 _q¹∗

≥

(An)^p2(Bn)^r(An)^p2 _q¹∗

, and it follows that

|Tⁿ|^2(p+r)nq∗ ≥

|Tⁿ|^pn|(Tⁿ)^∗|^2rn|Tⁿ|^np_q¹∗

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

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)^∗|^rn|Tⁿ|^2pn|(Tⁿ)^∗|^nr1_q

≥ |(Tⁿ)^∗|^2(p+r)nq

holds by Theorem D and the Löwner-Heinz theorem, so thatTⁿis a classwF(_n^p,_n^r, q)operator.

3. ANEXAMPLE

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

Theorem 3.1. LetAandBbe 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, and T = U D. Then the following assertions hold.

(1) IfT is a classwF(p, r, q)operator for 1≥ p > 0,1 ≥ r ≥ 0, q ≥ 1andrq ≤ 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 3.2. Noting that Theorem 3.1 holds 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 eachr≥0,

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

and

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

hold forp≥0andq ≥1with(1 +r)q≥p+r.

p

(1,0) q (0,−r)

(1,1)

q= 1 p=q

(1 +r)q =p+r

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

Proof of Theorem 3.1. By simple calculations, we have

|T|² =





















 . ..

B B

(A) A

A . ..





















 ,

|T^∗|² =





















 . ..

B B

(B) A

A . ..





















 ,

therefore

|T^∗|^r|T|^2p|T^∗|^r =





















 . ..

B^p+r

B^p+r

(B^r2A^pB^r2) A^p+r

A^p+r . ..























and

|T|^p|T^∗|^2r|T|^p =





















 . ..

B^p+r

B^p+r

(A^p2B^rA^p2) A^p+r

A^p+r . ..





















 ,

thus the following hold forn ≥2

T^n∗Tⁿ =









































 . ..

Bⁿ Bⁿ

Bⁿ⁻¹² ABⁿ⁻¹² . ..

B^j2A^n−jB^2j . ..

B¹²Aⁿ⁻¹B¹²

(Aⁿ) Aⁿ

. ..











































and

TⁿT^n∗ =









































 . ..

Bⁿ

(Bⁿ)

A¹²Bⁿ⁻¹A¹² . ..

A^j2B^n−jA^2j . ..

Aⁿ⁻¹² BAⁿ⁻¹² Aⁿ

Aⁿ . ..











































Proof of (1). T is a classwF(p, r, q)operator is equivalent to the following (B^r2A^pB^r2)^1q ≥B^p+rq and A^p+rq∗ ≥(A^p2B^rA^p2)^q1∗, Tⁿbelongs to classwF(_n^p,_n^r, q)is equivalent to the following (3.1) and (3.2).

(3.1)























(B^r2(B^2jA^n−jB^j2)^npB^r2)^1q ≥B^p+rq

(B^r2A^pB^r2)^1q ≥B^p+rq

A^j2B^n−jA^j2_2n^r A^p

A^j2B^n−jA^j2_2n^{r 1}_q

≥

A^j2B^n−jA^j2p+r_nq where j = 1,2, ..., n−1.

(3.2)























B^j2A^n−jB^j2_2n^p B^r

B^2jA^n−jB^j2_q¹∗

≥

B^j2A^n−jB^j2p+r_nq∗

A^p+rq∗ ≥(A^p2B^rA^p2)^q1∗

A^p+rq∗ ≥

A^p2

A^j2B^n−jA^2j_n^r A^p2

_q¹∗

where j = 1,2, ..., n−1.

We only prove (3.1) because of Theorem D.

Step 1. To show

B^r2

B^j2A^n−jB^j2p_n B^r2

¹_q

≥B^p+rq forj = 1,2, ..., n−1.

In fact, T is a class wF(p, r, q)operator for1 ≥ p > 0,1 ≥ r ≥ 0,q ≥ 1andrq ≤ p+r impliesT belongs to classwF

j, n−j,_δ+jⁿ

,whereδ = ^p+r_q −rby Theorem G and Theorem D, thus

B^j2A^n−jB^j2δ+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 Theorem H to

B^j2A^n−jB^2jδ+j_n

andB^δ+j for

1 + _δ+j^r

q ≥ _δ+j^p +_δ+j^r . Step 2. To show

A^2jB^n−jA^j2_2n^r A^p

A^j2B^n−jA^j2_2n^{r 1}_q

≥

A^j2B^n−jA^j2p+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−jA^{j2 j−δ}_n

,

this implies thatA^j ≥

A^2jB^n−jA^j2_n^j

by Theorem D. Therefore the assertion holds by apply- ing (i) of Theorem H toA^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.

We are indebted to Professor K. Tanahashi for a fruitful correspondence and the referee for his valuable advice and suggestions, especially for the improvement of Theorem 2.1.

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