#### Abstract

Let be a positive integer, and an operator is called a class operator if and -paranormal operator if for every unit vector , which are common generalizations of class and paranormal, respectively. In this paper, firstly we consider the tensor products for class operators, giving a necessary and sufficient condition for to be a class operator when and are both non-zero operators; secondly we consider the properties for -paranormal operators, showing that a -paranormal contraction is the direct sum of a unitary and a completely non-unitary contraction.

#### 1. Introduction

Let be a separable complex Hilbert space and let be the set of complex numbers. Let denote the -algebra of all bounded linear operators acting on . If , we will write and for the null space and range of , respectively. Also let , and let , denote the spectrum, point spectrum of . Let be the ascent of , that is, the smallest nonnegative integer such that . If such integer does not exist, we put . Analogously, let be the descent of , that is, the smallest nonnegative integer such that , and if such integer does not exist, we put . An operator is called upper (lower, resp.) semi-Fredholm if is closed and (, resp.). If is either an upper semi-Fredholm operator or a lower semi-Fredholm operator, then is called a semi-Fredholm operator, and the index of a semi-Fredholm operator , denoted by , is given by the integer . If both and are finite, then is called a Fredholm operator. An operator is called Weyl if it is Fredholm of index zero and Browder if it is Fredholm of finite ascent and descent. The essential spectrum , the Weyl spectrum , and the Browder spectrum of are defined by is not Fredholm}, is not Weyl}, and is not Browder.

Let , be complex Hilbert spaces and the tensor product of , , that is, the completion of the algebraic tensor product of , with the inner product for , , , . Let and . denotes the tensor product of and ; that is, for , .

A contraction is an operator such that ; equivalently, for every . A contraction is said to be a proper contraction if for every nonzero . A strict contraction is an operator such that . 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 is of class if when for every (i.e., is a strongly stable contraction) and it is said to be of class if for every nonzero . Classes and are defined by considering instead of and we define the class for , , 1 by . An isometry is a contraction for which for every .

Recall that is called -hyponormal for if [1]; when , is called hyponormal. And is called paranormal if for all [2, 3]. And is called normaloid if for all (equivalently, , the spectral radius of ). In order to discuss the relations between paranormal operators and -hyponormal and log-hyponormal operators ( is invertible and ), Furuta et al. [4] introduced a very interesting class of operators: class defined by , where which is called the absolute value of , and they showed that class is a subclass of paranormal and contains -hyponormal and log-hyponormal operators. Recently Yuan and Gao [5] introduced class (i.e., ) operators and -paranormal operators (i.e., for every unit vector ) for some positive integer .

For more interesting properties on class and -paranormal operators, see [5–8].

In general, the following implications hold:

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

#### 2. Tensor Products for Class Operators

Let denote the tensor product on the product space for nonzero and . The operation of taking tensor products preserves many properties of and , 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 is normal if and only if and are normal [10, 11]; however, there exist paranormal operators and such that is not paranormal [12]. Duggal [13] showed that for nonzero and , is -hyponormal if and only if , are -hyponormal. This result was extended to -quasihyponormal operators, class operators, log-hyponormal operators, and class operators (, , ) in [14–16], respectively. The following theorem gives a necessary and sufficient condition for to be a class operator when and are both nonzero operators.

Theorem 1. *Let and be nonzero operators. Then is a class operator if and only if and are class operators.*

*Proof. *It is clear that is a class operator if and only if
Therefore, the sufficiency is clear.

Conversely, suppose that is a class operator. Let and be arbitrary. Then we have
On the contrary, assume that is not a class operator; then there exists such that

From (3), we have
for all ; that is,
for all . Therefore, is a class operator. We have

So we have
for all by (6). By (8), we have

Since self-adjoint operators are normaloid, we have

Hence, we have

By (9) and (11), we have

This implies that . This contradicts the assumption . Hence must be a class operator. A similar argument shows that is also a class operator. The proof is complete.

#### 3. On -Paranormal Operators

An operator is said to have the single valued extension property (SVEP) at if, for every open neighborhood of , the only function such that on is , where means the space of all analytic functions on . When has SVEP at each , say that has SVEP.

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

Theorem 2 (see [19]). *Let be a contraction of -paranormal operators for a positive integer . Then is the direct sum of a unitary and a completely nonunitary contraction.*

*Proof. *If is a contraction, then the sequence is a decreasing sequence of self-adjoint operators, converging strongly to a contraction. Let . is self-adjoint and and . By [20] we have that there exists an isometry : such that on and for every . can be extended to a bounded linear operator on ; we still denote it by . Let , . Then for all nonnegative integers ,
So we have, for all , . The sequence is a bounded above increasing sequence. In the following, we will prove that if is -paranormal for a positive integer , then is a projection. Firstly we prove that is a constant sequence. Suppose that is a -paranormal operator for a positive integer . Then, for all and nonzero ,
so we have
Hence,
Putting , we have that
where and as . Suppose that there exists an integer such that ; then , and we have that , for all by an induction argument. This is contradictory with the fact that as . Consequently, we have that for all , which implies that for all . This means that for all . So we have that on , and so on . Therefore, we have that on . Hence is a projection. By [21], we have that if is a projection, then has a decomposition:
where is unitary and the completely nonunitary part of is the direct sum of backward unilateral shift and a -contraction . We will prove that is missing from the direct sum. It is well known that an operator has SVEP at a point if and only if and have SVEP at the point . Since -paranormal operators have SVEP by [6, Corollary 3.4], it follows that if is present in the direct sum of , 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, . The proof is complete.

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

Theorem 3. *Let be a contraction of -paranormal operators for a positive integer . If has no nontrivial invariant subspace, then is a proper contraction.*

*Proof. *Suppose that is a -paranormal operator, then for all . By [22, Theorem 3.6], we have that
Put , which is a subspace of . In the following, we will show that is an invariant subspace of . For every , if is a -paranormal operator, we have

By (20) we have . So we have that
That is, is an invariant subspace of . Now suppose that is a contraction of -paranormal operators. If is a strict contract, then it is trivially a proper contraction. If is not a strict contraction (i.e., ) and has no nontrivial invariant subspace, then (actually, if , then is an isometry, and isometries have nontrivial invariant subspaces). Thus for every nonzero , , so is a proper contraction. The proof is complete.

Uchiyama [23] showed that if is paranormal and , then is compact and normal. Now we extend this result to -paranormal operators.

Theorem 4. *Let be a -paranormal operator for a positive integer and . Then is compact and normal.*

*Proof. *By [5, Theorem 2.1], we have that
where is the set of all isolated points which are eigenvalues of with finite multiplicities. This implies that is a finite set or a countable infinite set with 0 as its only accumulation point. Let , where whenever and is a nonincreasing sequence. By [8, Proposition 1], we have that is normaloid. So we have . By the general theory, implies . In fact,
Thus and . Therefore, is a reducing subspace of . Let be the orthogonal projection onto . Then on . Since is -paranormal and , we have that . By the same argument as above, is a finite dimensional reducing subspace of which is included in . Let be the orthogonal projection onto . Then on . By the same argument, each is a reducing subspace of and as . Here is the orthogonal projection onto and on . Hence is compact and normal because each is a finite rank orthogonal projection which satisfies whenever by [5, Lemma 2.5] and as . 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).