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
yuanjiangtao02@yahoo.com.cn
COLLEGE OFMATHEMATICS ANDINFORMATIONSCIENCE
HENANNORMALUNIVERSITY
XINXIANG453007, CHINA
yangchangsen117@yahoo.com.cn
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 ≥ 0ensuresAα ≥ Bα 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(Xk−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+rr
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+rr )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(np,nr)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(np,nr, q)operator.
Theorem C ([25]). Let1 ≥ p > 0,1 ≥ r ≥ 0; q ≥ 1when r = 0 and p+rr ≥ q ≥ 1when r >0. IfT is a classwF(p, r, q)operator, thenTnis a classwF(np,nr, q)operator.
Here we generalize them to the following.
Theorem 2.1. Let1 ≥ p > 0, 1 ≥ r > 0; q > p+rr . IfT is a classwF(p, r, q)operator such thatN(T)⊂N(T∗), thenTnis a classwF(pn,nr, 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) Br2ApBr2p+rr
≥Br ⇒ Ap2BrAp2p+rp
≤Ap. (2) Ap2BrAp2p+rp
≤Ap andN(A)⊂N(B)⇒ Br2ApBr2p+rr
≥Br.
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,pδ0+r0
0+r0
operator forp0 >0,r0 ≥0and−r0 <
δ0 ≤p0. ThenT is a classwF
p, r,δp+r
0+r
operator forp≥p0andr≥r0.
Proposition A ([25]). LetA, B ≥0;1≥p >0,1 ≥r > 0; p+rr ≥q ≥1. Then the following assertions hold.
(1) If Br2ApBr21q
≥Bp+rq andB ≥C, then Cr2ApCr21q
≥Cp+rq . (2) IfBp+rq ≥ Br2CpBr21q
,A≥B and the condition
(*) if lim
n→∞B12xn = 0 and lim
n→∞A12xnexists, then lim
n→∞A12xn= 0 holds for any sequence of vectors{xn}, thenAp+rq ≥ Ar2CpAr21q
.
Proof of Theorem 2.1. Putδ = p+rq −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) An≥ · · · ≥A2 ≥A1 and B1 ≥B2 ≥ · · · ≥Bn.
Meanwhile,AnandA1 satisfy the following for any sequence of vectors{xm}(see [24]) if lim
m→∞A
1 2
1xm = 0 and lim
m→∞A
1
n2xmexists, then lim
m→∞A
1
n2xm = 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|npq1∗
. 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)∗|nr1q
≥ |(Tn)∗|2(p+r)nq
holds by Theorem D and the Löwner-Heinz theorem, so thatTnis a classwF(np,nr, 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=−∞Hk, whereHk ∼=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(pn,nr, 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(np,nr, 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,
(Br2ApBr2)1q ≥(Br2BpBr2)1q (i)
and
(Ar2ApAr2)1q ≥(Ar2BpAr2)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 =
. ..
Bp+r
Bp+r
(Br2ApBr2) Ap+r
Ap+r . ..
and
|T|p|T∗|2r|T|p =
. ..
Bp+r
Bp+r
(Ap2BrAp2) Ap+r
Ap+r . ..
,
thus the following hold forn ≥2
Tn∗Tn =
. ..
Bn Bn
Bn−12 ABn−12 . ..
Bj2An−jB2j . ..
B12An−1B12
(An) An
. ..
and
TnTn∗ =
. ..
Bn
(Bn)
A12Bn−1A12 . ..
Aj2Bn−jA2j . ..
An−12 BAn−12 An
An . ..
Proof of (1). T is a classwF(p, r, q)operator is equivalent to the following (Br2ApBr2)1q ≥Bp+rq and Ap+rq∗ ≥(Ap2BrAp2)q1∗, Tnbelongs to classwF(np,nr, q)is equivalent to the following (3.1) and (3.2).
(3.1)
(Br2(B2jAn−jBj2)npBr2)1q ≥Bp+rq
(Br2ApBr2)1q ≥Bp+rq
Aj2Bn−jAj22nr Ap
Aj2Bn−jAj22nr 1q
≥
Aj2Bn−jAj2p+rnq where j = 1,2, ..., n−1.
(3.2)
Bj2An−jBj22np Br
B2jAn−jBj2q1∗
≥
Bj2An−jBj2p+rnq∗
Ap+rq∗ ≥(Ap2BrAp2)q1∗
Ap+rq∗ ≥
Ap2
Aj2Bn−jA2jnr Ap2
q1∗
where j = 1,2, ..., n−1.
We only prove (3.1) because of Theorem D.
Step 1. To show
Br2
Bj2An−jBj2pn Br2
1q
≥Bp+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+rq −rby Theorem G and Theorem D, thus
Bj2An−jBj2δ+jn
≥Bδ+j and An−j−δ≥
An−j2 BjAn−j2 n−j−δn
Therefore the assertion holds by applying (i) of Theorem H to
Bj2An−jB2jδ+jn
andBδ+j for
1 + δ+jr
q ≥ δ+jp +δ+jr . Step 2. To show
A2jBn−jAj22nr Ap
Aj2Bn−jAj22nr 1q
≥
Aj2Bn−jAj2p+rnq forj = 1,2, ..., n−1.
In fact, similar to Step 1, the following hold
Bn−j2 AjBn−j2 δ+n−jn
≥Bδ+n−j and Aj−δ ≥
Aj2Bn−jAj2 j−δn
,
this implies thatAj ≥
A2jBn−jAj2nj
by Theorem D. Therefore the assertion holds by apply- ing (i) of Theorem H toAj and
Aj2Bn−jAj2nj
for(1 + rj)q≥ pj +rj.
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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