# On Properties of ClassA(n) and n-Paranormal Operators

Authors: Xiaochun Li; Fugen Gao
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

Letn be a positive integer, and an operator T∈B(ℋ) 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∈ℋ,  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 T⊗S 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.


---

## Body

## 1. Introduction

Letℋ be a separable complex Hilbert space and let 𝒞 be the set of complex numbers. Let B(ℋ) denote the C*-algebra of all bounded linear operators acting on ℋ. If T∈B(ℋ), we will write ker⁡T and ranT for the null space and range of T, respectively. Also let α(T)=dim⁡⁡ker⁡T, β(T)=dim⁡⁡ker⁡T* 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 ker⁡Tp=ker⁡Tp+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 T∈B(ℋ) is called upper (lower, resp.) semi-Fredholm if ranT is closed and α(T)<∞ (β(T)<∞, resp.). If T∈B(ℋ) 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 T∈B(ℋ), 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 T∈B(ℋ) 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 T∈B(ℋ) are defined by σe(T)={λ∈𝒞:T-λ is not Fredholm}, σw(T)={λ∈𝒞:T-λ is not Weyl}, and σb(T)={λ∈𝒞:T-λ is not Browder}.Letℋ, 𝒦 be complex Hilbert spaces and ℋ⊗𝒦 the tensor product of ℋ, 𝒦, that is, the completion of the algebraic tensor product of ℋ, 𝒦 with the inner product 〈x1⊗y1,x2⊗y2〉=〈x1,x2〉〈y1,y2〉 for x1, x2∈ℋ, y1, y2∈𝒦. Let T∈B(ℋ) and S∈B(𝒦). T⊗S∈B(ℋ⊗𝒦) denotes the tensor product of T and S; that is, (T⊗S)(x⊗y)=Tx⊗Sy for x∈ℋ, y∈𝒦.A contraction is an operatorT such that ∥T∥≤1; equivalently, ∥Tx∥≤∥x∥ for every x∈ℋ. A contraction T is said to be a proper contraction if ∥Tx∥<∥x∥ for every nonzero x∈ℋ. 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 ∥Tnx∥→0 when n→∞ for every x∈ℋ (i.e., T is a strongly stable contraction) and it is said to be of class C1. if lim⁡n→∞∥Tnx∥>0 for every nonzero x∈ℋ. 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∈ℋ.Recall thatT∈B(ℋ) is called p-hyponormal for p>0 if (T*T)p-(TT*)p≥0 [1]; when p=1, T is called hyponormal. And T is called paranormal if ∥Tx∥2≤∥T2x∥∥x∥ for all x∈ℋ [2, 3]. And T is called normaloid if ∥Tn∥=∥T∥n for all n∈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 log⁡T*T≥log⁡TT*), Furuta et al. [4] introduced a very interesting class of operators: class A defined by |T2|-|T|2≥0, 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 [5] introduced class A(n) (i.e., |T1+n|2/(1+n)≥|T|2) operators and n-paranormal operators (i.e., ∥T1+nx∥1/(1+n)≥∥Tx∥ for every unit vector x∈ℋ) for some positive integer n.For more interesting properties on classA(n) and n-paranormal operators, see [5–8].In general, the following implications hold:(1)p-hyponormal⊆classA⊆paranormal⊆n-paranormal,p-hyponormal⊆classA⊆classA(n)⊆n-paranormal.In this paper, firstly we consider the tensor products for classA(n) operators, giving a necessary and sufficient condition for T⊗S 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 ClassA(n) Operators

LetT⊗S denote the tensor product on the product space ℋ⊗𝒦 for nonzero T∈B(ℋ) and S∈B(𝒦). The operation of taking tensor products T⊗S preserves many properties of T∈B(ℋ) and S∈B(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 T⊗S is normal if and only if T and S are normal [10, 11]; however, there exist paranormal operators T∈B(ℋ) and S∈B(𝒦) such that T⊗S is not paranormal [12]. Duggal [13] showed that for nonzero T∈B(ℋ) and S∈B(𝒦), T⊗S 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 [14–16], respectively. The following theorem gives a necessary and sufficient condition for T⊗S to be a class A(n) operator when T and S are both nonzero operators.Theorem 1.
LetT∈B(ℋ) and S∈B(𝒦) be nonzero operators. Then T⊗S∈B(ℋ⊗𝒦) is a class A(n) operator if and only if T and S are class A(n) operators.Proof.
It is clear thatT⊗S is a class A(n) operator if and only if
(2)|(T⊗S)1+n|2/(1+n)≥|T⊗S|2⟺|T1+n⊗S1+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 thatT⊗S is a class A(n) operator. Let x∈ℋ and y∈𝒦 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,y〉≥0.
On the contrary, assume that T is not a class A(n) operator; then there exists x0∈ℋ 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,y〉≥0
for all y∈𝒦; that is,
(6)(α+β)〈|S1+n|2/(1+n)y,y〉≥β〈|S|2y,y〉
for all y∈𝒦. Therefore, S is a class A(n) operator. We have
(7)〈|S|2y,y〉=∥Sy∥2,〈|S1+n|2/(1+n)y,y〉=∥|S1+n|1/(1+n)y∥2.
So we have(8)(α+β)∥|S1+n|1/(1+n)y∥2≥β∥Sy∥2
for all y∈𝒦 by (6). By (8), we have
(9)(α+β)∥|S1+n|1/(1+n)∥2≥β∥S∥2.
Since self-adjoint operators are normaloid, we have(10)∥|S1+n|1/(1+n)∥1+n=∥(|S1+n|1/(1+n))1+n∥=∥S1+n∥≤∥S∥1+n.
Hence, we have(11)∥|S1+n|1/(1+n)∥≤∥S∥.
By (9) and (11), we have
(12)(α+β)∥S∥2≥β∥S∥2.
This implies thatS=0. This contradicts the assumption S≠0. 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. Onn-Paranormal Operators

An operatorT∈B(ℋ) is said to have the single valued extension property (SVEP) at λ∈ℂ if, for every open neighborhood 𝒢 of λ, the only function f∈H(𝒢) such that (T-μ)f(μ)=0 on G is 0∈H(𝒢), where H(𝒢) means the space of all analytic functions on G. When T has SVEP at each λ∈ℂ, say that T has SVEP.In the following, we consider the properties ofn-paranormal operators. References [17, 18] showed that paranormal contractions and *-paranormal contractions in B(ℋ) 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 [19]).
LetT 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.
IfT is a contraction, then the sequence {TkT*k} is a decreasing sequence of self-adjoint operators, converging strongly to a contraction. Let A=(lim⁡k→∞TkT*k)1/2. A is self-adjoint and 0≤A≤I and TA2T*=A2. By [20] we have that there exists an isometry V: ran(A)¯→ran(A)¯ such that VA=AT* on ran(A)¯ and ∥AVnx∥→∥x∥ for every x∈ran(A)¯. V can be extended to a bounded linear operator on ℋ; we still denote it by V. Let xk=AVkx, k∈ℕ∪{0}. Then for all nonnegative integers m,
(13)Tmxm+k=TmAVk+mx=AV*mVk+mx=AVkx=xk.
So we have, for all m≤k, 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  k≥1 and nonzero x∈ran(A)¯,
(14)∥xk∥2=∥Txk+1∥2≤∥T1+nxk+1∥2/(1+n)∥xk+1∥2n/(1+n)=∥xk+1-(1+n)∥2/(1+n)∥xk+1∥2n/(1+n)=∥xk-n∥2/(1+n)∥xk+1∥2n/(1+n),
so we have
(15)∥xk∥≤∥xk-n∥1/(n+1)∥xk+1∥n/(n+1)≤1n+1(∥xk-n∥+n∥xk+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+1≥bk+bk+1+⋯+bk-n+1,
where bk≥0 and bk→0 as k→∞. Suppose that there exists an integer i≥1 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 bk→0 as k→∞. Consequently, we have that bk=0 for all k, which implies that ∥xk-1∥=∥xk∥ for all k≥1. This means that for all x∈ran(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 ℋ=ran(A)¯⊕ker⁡(A). Hence A is a projection. By [21], we have that if A is a projection, then T has a decomposition:
(18)T=Tu⊕Tc,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=B1⊕B2 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=Tu⊕T0. The proof is complete.In the following, we give a sufficient condition for an-paranormal contraction to be proper.Theorem 3.
LetT 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 thatT is a n-paranormal operator, then ∥T1+nx∥∥x∥n≥∥Tx∥n+1 for all x∈ℋ. By [22, Theorem 3.6], we have that
(19)T*Tx=∥T∥2xif and only if∥Tx∥=∥T∥∥x∥.
Put 𝒰={x∈ℋ:∥Tx∥=∥T∥∥x∥}=ker⁡(|T|2-∥T∥2), which is a subspace of ℋ. In the following, we will show that 𝒰 is an invariant subspace of T. For every x∈𝒰, if T is a n-paranormal operator, we have
(20)∥T∥n+1∥x∥n+1=∥Tx∥1+n≤∥T1+nx∥∥x∥n≤∥T∥n-1∥T2x∥∥x∥n.
By (20) we have ∥T2x∥≥∥T∥2∥x∥. So we have that
(21)∥T(Tx)∥=∥T∥∥Tx∥.
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∈ℋ:∥Tx∥=∥x∥}={0} (actually, if 𝒰=ℋ, then T is an isometry, and isometries have nontrivial invariant subspaces). Thus for every nonzero x∈ℋ, ∥Tx∥<∥x∥, so T is a proper contraction. The proof is complete.Uchiyama [23] 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.
LetT 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 λn≠λm whenever n≠m 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)∥(∥T∥2-T*T)1/2x∥2=∥T∥2∥x∥2-∥Tx∥2=∥T∥2∥x∥2-∥λ1x∥2=0.
Thus λ1T*x=T*Tx=∥T∥2x=|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=λ1⊕T1 on ℋ=E1ℋ⊕(I-E1)ℋ. Since T1 is n-paranormal and σp(T)=σp(T1)∪{λ1}, we have that λ2∈σp(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)ℋ. Let E2 be the orthogonal projection onto ker⁡(T-λ2). Then T=λ1E1⊕λ2E2⊕T2 on ℋ=E1ℋ⊕E2ℋ⊕(I-E1-E2)ℋ. 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 ℋ=(λ1E1⨁k=1nEkℋ)⊕(I-Σk=1nEk)ℋ. Hence T=⨁k=1∞λkEk is compact and normal because each Ek is a finite rank orthogonal projection which satisfies EkEl=0 whenever k≠l by [5, Lemma 2.5] and λn→0 as n→∞. The proof is complete.



---

*Source: 629061-2014-03-12.xml*