AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 629061 10.1155/2014/629061 629061 Research Article On Properties of Class A(n) and n-Paranormal Operators Li Xiaochun http://orcid.org/0000-0002-4817-5227 Gao Fugen Yang Changsen College of Mathematics and Information Science Henan Normal University Xinxiang Henan 453007 China henannu.edu.cn 2014 1232014 2014 26 12 2013 06 02 2014 12 3 2014 2014 Copyright © 2014 Xiaochun Li and Fugen Gao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Let n be a positive integer, and an operator TB() is called a class A(n) operator if T1+n2/1+n|T|2 and n-paranormal operator if T1+nx1/1+n||Tx|| for every unit vector x, which are common generalizations of class A and paranormal, respectively. In this paper, firstly we consider the tensor products for class A(n) operators, giving a necessary and sufficient condition for TS to be a class A(n) operator when T and S are both non-zero operators; secondly we consider the properties for n-paranormal operators, showing that a n-paranormal contraction is the direct sum of a unitary and a C.0 completely non-unitary contraction.

1. Introduction

Let be a separable complex Hilbert space and let 𝒞 be the set of complex numbers. Let B() denote the C*-algebra of all bounded linear operators acting on . If TB(), we will write kerT and ranT for the null space and range of T, respectively. Also let α(T)=dimkerT, β(T)=dimkerT* and let σ(T), σp(T) denote the spectrum, point spectrum of T. Let p=p(T) be the ascent of T, that is, the smallest nonnegative integer p such that kerTp=kerTp+1. If such integer does not exist, we put p(T)=. Analogously, let q=q(T) be the descent of T, that is, the smallest nonnegative integer q such that ranTq=ranTq+1, and if such integer does not exist, we put q(T)=. An operator TB() is called upper (lower, resp.) semi-Fredholm if ranT is closed and α(T)< (β(T)<, resp.). If TB() is either an upper semi-Fredholm operator or a lower semi-Fredholm operator, then T is called a semi-Fredholm operator, and the index of a semi-Fredholm operator TB(), denoted by ind(T), is given by the integer ind(T)=α(T)-β(T). If both α(T) and β(T) are finite, then T is called a Fredholm operator. An operator TB() is called Weyl if it is Fredholm of index zero and Browder if it is Fredholm of finite ascent and descent. The essential spectrum σe(T), the Weyl spectrum σw(T), and the Browder spectrum σb(T) of TB() are defined by σe(T)={λ𝒞:T-λ is not Fredholm}, σw(T)={λ𝒞:T-λ is not Weyl}, and σb(T)={λ𝒞:T-λ is not Browder}.

Let , 𝒦 be complex Hilbert spaces and 𝒦 the tensor product of , 𝒦, that is, the completion of the algebraic tensor product of , 𝒦 with the inner product x1y1,x2y2=x1,x2y1,y2 for x1, x2, y1, y2𝒦. Let TB() and SB(𝒦). TSB(𝒦) denotes the tensor product of T and S; that is, (TS)(xy)=TxSy for x, y𝒦.

A contraction is an operator T such that T1; equivalently, Txx for every x. A contraction T is said to be a proper contraction if Tx<x for every nonzero x. A strict contraction is an operator T such that T<1. A strict contraction is a proper contraction, but a proper contraction is not necessarily a strict contraction, although the concepts of strict and proper contraction coincide for compact operators. A contraction T is of class C0. if Tnx0 when n for every x (i.e., T is a strongly stable contraction) and it is said to be of class C1. if limnTnx>0 for every nonzero x. Classes C.0 and C.1 are defined by considering T* instead of T and we define the class Cαβ for α, β=0, 1 by Cαβ=Cα.C.β. An isometry is a contraction for which Tx=x for every x.

Recall that TB() is called p-hyponormal for p>0 if (T*T)p-(TT*)p0 ; when p=1, T is called hyponormal. And T is called paranormal if Tx2T2xx for all x [2, 3]. And T is called normaloid if Tn=Tn for all n (equivalently, T=r(T), the spectral radius of T). In order to discuss the relations between paranormal operators and p-hyponormal and log-hyponormal operators (T is invertible and logT*TlogTT*), Furuta et al.  introduced a very interesting class of operators: class A defined by |T2|-|T|20, where |T|=(T*T)1/2 which is called the absolute value of T, and they showed that class A is a subclass of paranormal and contains p-hyponormal and log-hyponormal operators. Recently Yuan and Gao  introduced class A(n) (i.e., |T1+n|2/(1+n)|T|2) operators and n-paranormal operators (i.e., T1+nx1/(1+n)Tx for every unit vector x) for some positive integer n.

For more interesting properties on class A(n) and n-paranormal operators, see .

In general, the following implications hold: (1)p-hyponormalclassAparanormaln-paranormal,p-hyponormalclassAclassA(n)n-paranormal.

In this paper, firstly we consider the tensor products for class A(n) operators, giving a necessary and sufficient condition for TS to be a class A(n) operator when T and S are both nonzero operators; secondly we consider the properties for n-paranormal operators, showing that a n-paranormal contraction is the direct sum of a unitary and a   C.0 completely nonunitary contraction.

2. Tensor Products for Class <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M152"><mml:mi>A</mml:mi><mml:mo mathvariant="bold">(</mml:mo><mml:mi>n</mml:mi><mml:mo mathvariant="bold">)</mml:mo></mml:math></inline-formula> Operators

Let TS denote the tensor product on the product space 𝒦 for nonzero TB() and SB(𝒦). The operation of taking tensor products TS preserves many properties of TB() and SB(𝒦), but it was not always this way. For example, the normaloid property is invariant under tensor products, the spectraloid property is not (see [9, pp. 623 and 631]), and TS is normal if and only if T and S are normal [10, 11]; however, there exist paranormal operators TB() and SB(𝒦) such that TS is not paranormal . Duggal  showed that for nonzero TB() and SB(𝒦), TS is p-hyponormal if and only if T, S are p-hyponormal. This result was extended to p-quasihyponormal operators, class A operators, log-hyponormal operators, and class A(s,t) operators ((|T*|t|T|2s|T*|t)t/(s+t)|T*|2t, s, t>0) in , respectively. The following theorem gives a necessary and sufficient condition for TS to be a class A(n) operator when T and S are both nonzero operators.

Theorem 1.

Let TB() and SB(𝒦) be nonzero operators. Then TSB(𝒦) is a class A(n) operator if and only if T and S are class A(n) operators.

Proof.

It is clear that TS is a class A(n) operator if and only if (2)|(TS)1+n|2/(1+n)|TS|2|T1+nS1+n|2/(1+n)|T|2|S|2(|T1+n|2/(1+n)-|T|2)|S1+n|2/(1+n)+|T|2(|S1+n|2/(1+n)-|S|2)0. Therefore, the sufficiency is clear.

Conversely, suppose that TS is a class A(n) operator. Let x and y𝒦 be arbitrary. Then we have (3)(|T1+n|2/(1+n)-|T|2)x,x|S1+n|2/(1+n)y,y+|T|2x,x(|S1+n|2/(1+n)-|S|2)y,y0. On the contrary, assume that T is not a class A(n) operator; then there exists x0 such that (4)(|T1+n|2/(1+n)-|T|2)x0,x0=α<0,|T|2x0,x0=β>0.

From (3), we have (5)α|S1+n|2/(1+n)y,y+β(|S1+n|2/(1+n)-|S|2)y,y0 for all y𝒦; that is, (6)(α+β)|S1+n|2/(1+n)y,yβ|S|2y,y for all y𝒦. Therefore, S is a class A(n) operator. We have (7)|S|2y,y=Sy2,|S1+n|2/(1+n)y,y=|S1+n|1/(1+n)y2.

So we have (8)(α+β)|S1+n|1/(1+n)y2βSy2 for all y𝒦 by (6). By (8), we have (9)(α+β)|S1+n|1/(1+n)2βS2.

Since self-adjoint operators are normaloid, we have (10)|S1+n|1/(1+n)1+n=(|S1+n|1/(1+n))1+n=S1+nS1+n.

Hence, we have (11)|S1+n|1/(1+n)S.

By (9) and (11), we have (12)(α+β)S2βS2.

This implies that S=0. This contradicts the assumption S0. Hence T must be a class A(n) operator. A similar argument shows that S is also a class A(n) operator. The proof is complete.

3. On <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M221"><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:math></inline-formula>-Paranormal Operators

An operator TB() is said to have the single valued extension property (SVEP) at λ if, for every open neighborhood 𝒢 of λ, the only function fH(𝒢) such that (T-μ)f(μ)=0 on G is 0H(𝒢), where H(𝒢) means the space of all analytic functions on G. When T has SVEP at each λ, say that T has SVEP.

In the following, we consider the properties of n-paranormal operators. References [17, 18] showed that paranormal contractions and *-paranormal contractions in B() are the direct sum of a unitary and a C.0 contraction. In the following theorem, we extend this result to n-paranormal operators.

Theorem 2 (see [<xref ref-type="bibr" rid="B17">19</xref>]).

Let T be a contraction of n-paranormal operators for a positive integer n. Then T is the direct sum of a unitary and a C.0 completely nonunitary contraction.

Proof.

If T is a contraction, then the sequence {TkT*k} is a decreasing sequence of self-adjoint operators, converging strongly to a contraction. Let A=(limkTkT*k)1/2. A is self-adjoint and 0AI and TA2T*=A2. By  we have that there exists an isometry V: ran(A)¯ran(A)¯ such that VA=AT* on ran(A)¯ and AVnxx for every xran(A)¯. V can be extended to a bounded linear operator on ; we still denote it by V. Let xk=AVkx, k{0}. Then for all nonnegative integers m, (13)Tmxm+k=TmAVk+mx=AV*mVk+mx=AVkx=xk. So we have, for all mk, Tmxk=xk-m. The sequence {xn} is a bounded above increasing sequence. In the following, we will prove that if T is n-paranormal for a positive integer n, then A is a projection. Firstly we prove that {xk} is a constant sequence. Suppose that T is a n-paranormal operator for a positive integer n. Then, for all  k1 and nonzero xran(A)¯, (14)xk2=Txk+12T1+nxk+12/(1+n)xk+12n/(1+n)=xk+1-(1+n)2/(1+n)xk+12n/(1+n)=xk-n2/(1+n)xk+12n/(1+n), so we have (15)xkxk-n1/(n+1)xk+1n/(n+1)1n+1(xk-n+nxk+1). Hence, (16)n(xk+1-xk)xk-xk-n=(xk-xk-1)+(xk-1-xk-2)++(xk-n+1-xk-n). Putting bk=xk-xk-1, we have that (17)nbk+1bk+bk+1++bk-n+1, where bk0 and bk0 as k. Suppose that there exists an integer i1 such that bi>0; then bi+1(bi/n)>0, and we have that bk(bi/n)>0, for all k>i by an induction argument. This is contradictory with the fact that bk0 as k. Consequently, we have that bk=0 for all k, which implies that xk-1=xk for all k1. This means that for all xran(A)¯AVkx=Ax=x. So we have that A2=I on ran(A)¯, and so A=I on ran(A)¯. Therefore, we have that A=(I000) on =ran(A)¯ker(A). Hence A is a projection. By , we have that if A is a projection, then T has a decomposition: (18)T=TuTc,Tc=S*T0, where Tu is unitary and the completely nonunitary part Tc of T is the direct sum of backward unilateral shift S* and a C.0-contraction T0. We will prove that S* is missing from the direct sum. It is well known that an operator B=B1B2 has SVEP at a point λ if and only if B1 and B2 have SVEP at the point λ. Since n-paranormal operators have SVEP by [6, Corollary 3.4], it follows that if S* is present in the direct sum of T, then it has SVEP. This contradicts the fact that the backward unilateral shift does not have SVEP anywhere on its spectrum except for the boundary point of its spectrum. Therefore, T=TuT0. The proof is complete.

In the following, we give a sufficient condition for a n-paranormal contraction to be proper.

Theorem 3.

Let T be a contraction of n-paranormal operators for a positive integer n. If T has no nontrivial invariant subspace, then T is a proper contraction.

Proof.

Suppose that T is a n-paranormal operator, then T1+nxxnTxn+1 for all x. By [22, Theorem 3.6], we have that (19)T*Tx=T2xif and only if  Tx=Tx. Put 𝒰={x:Tx=Tx}=ker(|T|2-T2), which is a subspace of . In the following, we will show that 𝒰 is an invariant subspace of T. For every x𝒰, if T is a n-paranormal operator, we have (20)Tn+1xn+1=Tx1+nT1+nxxnTn-1T2xxn.

By (20) we have T2xT2x. So we have that (21)T(Tx)=TTx. That is, 𝒰 is an invariant subspace of T. Now suppose that T is a contraction of n-paranormal operators. If T is a strict contract, then it is trivially a proper contraction. If T is not a strict contraction (i.e., T=1) and T has no nontrivial invariant subspace, then 𝒰={x:Tx=x}={0} (actually, if 𝒰=, then T is an isometry, and isometries have nontrivial invariant subspaces). Thus for every nonzero x, Tx<x, so T is a proper contraction. The proof is complete.

Uchiyama  showed that if T is paranormal and w(T)=0, then T is compact and normal. Now we extend this result to n-paranormal operators.

Theorem 4.

Let T be a n-paranormal operator for a positive integer n and σw(T)={0}. Then T is compact and normal.

Proof.

By [5, Theorem 2.1], we have that (22)σ(T)w(T)=σ(T){0}π00(T), where π00(T) is the set of all isolated points which are eigenvalues of T with finite multiplicities. This implies that σ(T)w(T) is a finite set or a countable infinite set with 0 as its only accumulation point. Let σ(T){0}={λn}, where λnλm whenever nm and {|λn|} is a nonincreasing sequence. By [8, Proposition 1], we have that T is normaloid. So we have |λ1|=T. By the general theory, (T-λ1)x=0 implies (T-λ1)*x=0. In fact, (23)(T2-T*T)1/2x2=T2x2-Tx2=T2x2-λ1x2=0. Thus λ1T*x=T*Tx=T2x=|λ1|2x and T*x=λ1¯x. Therefore, ker(T-λ1) is a reducing subspace of T. Let E1 be the orthogonal projection onto ker(T-λ1). Then T=λ1T1 on =E1(I-E1). Since T1 is n-paranormal and σp(T)=σp(T1){λ1}, we have that λ2σp(T1). By the same argument as above, ker(T-λ2)=ker(T1-λ2) is a finite dimensional reducing subspace of T which is included in (I-E1). Let E2 be the orthogonal projection onto ker(T-λ2). Then T=λ1E1λ2E2T2 on =E1E2(I-E1-E2). By the same argument, each ker(T-λn) is a reducing subspace of T and T-k=1nλkEk=Tn=|λn+1|0 as n. Here Ek is the orthogonal projection onto ker(T-λk) and T=(k=1nλkEk)Tn on =(λ1E1k=1nEk)(I-Σk=1nEk). Hence T=k=1λkEk is compact and normal because each Ek is a finite rank orthogonal projection which satisfies EkEl=0 whenever kl by [5, Lemma 2.5] and λn0 as n. The proof is complete.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research is supported by the National Natural Science Foundation of China ((11301155) and (11271112)), the Natural Science Foundation of the Department of Education, Henan Province ((2011A110009) and (13B110077)), the Youth Science Foundation of Henan Normal University, and the new teachers Science Foundation of Henan Normal University (no. qd12102).

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