- Research Article
- Open Access

- Fugen Gao
^{1, 2}Email author and - Xiaochun Fang
^{1}

**2009**:921634

https://doi.org/10.1155/2009/921634

© F. Gao and X. Fang. 2009

**Received:**26 June 2009**Accepted:**10 November 2009**Published:**13 December 2009

## Abstract

An operator is called -quasiclass if for a positive integer , which is a common generalization of quasiclass . In this paper, firstly we prove some inequalities of this class of operators; secondly we prove that if is a -quasiclass operator, then is isoloid and has finite ascent for all complex number at last we consider the tensor product for -quasiclass operators.

## Keywords

- Positive Integer
- Tensor Product
- Orthonormal Basis
- Space Operator
- Orthogonal Projection

## 1. Introduction

Throughout this paper let be a separable complex Hilbert space with inner product . Let denote the -algebra of all bounded linear operators on .

Let and let be an isolated point of . Here denotes the spectrum of . Then there exists a small enough positive number such that

Let

After that many authors extended this result to many other classes of operators. Chō and Tanahashi [2] proved that (1.3) holds if is either -hyponormal or log-hyponormal. In the case , the result was further shown by Tanahashi and Uchiyama [3] to hold for -quasihyponormal operators, by Tanahashi et al. [4] to hold for -quasihyponormal operators and by Uchiyama and Tanahashi [5] and Uchiyama [6] for class A and paranormal operators. Here an operator is called -hyponormal for if , and log-hyponormal if is invertible and . An operator is called -quasihyponormal if , where and is a positive integer; especially, when , , and , is called -quasihyponormal, -quasihyponormal, and quasihyponormal, respectively. And an operator is called paranormal if for all ; normaloid if for all positive integers . -hyponormal, log-hyponormal, -quasihyponormal, -quasihyponormal, and paranormal operators were introduced by Aluthge [7], Tanahashi [8], S. C. Arora and P. Arora [9], Kim [10], and Furuta [11, 12], respectively.

In order to discuss the relations between paranormal and -hyponormal and log-hyponormal operators, Furuta et al. [13] introduced a very interesting class of bounded linear Hilbert space operators: class A defined by , where which is called the absolute value of and they showed that class A is a subclass of paranormal and contains -hyponormal and log-hyponormal operators. Class A operators have been studied by many researchers, for example, [5, 14–19].

Recently Jeon and Kim [20] introduced quasiclass A (i.e., ) operators as an extension of the notion of class A operators, and they also proved that (1.3) holds for this class of operators when . It is interesting to study whether Stampli's result holds for other larger classes of operators.

In [21], Tanahashi et al. considered an extension of quasi-class A operators, similar in spirit to the extension of the notion of -quasihyponormality to -quasihyponormality, and prove that (1.3) holds for this class of operators in the case .

Definition 1.1.

Remark 1.2.

In [21], this class of operators is called quasi-class (A, ).

It is clear that the class of quasi-class
A operators
the class of *k*-quasiclass A operators and

We show that the inclusion relation (1.5) is strict, by an example which appeared in [20].

Example 1.3.

Straightforward calculations show that is a -quasiclass A operator if and only if . So if and , then is a -quasiclass A operator, but not a -quasiclass A operator.

In this paper, firstly we consider some inequalities of -quasiclass A operators; secondly we prove that if is a -quasiclass A operator, then is isoloid and has finite ascent for all complex number ; at last we give a necessary and sufficient condition for to be a -quasiclass A operator when and are both non-zero operators.

## 2. Results

In the following lemma, Tanahashi, Jeon, Kim, and Uchiyama studied the matrix representation of a -quasiclass A operator with respect to the direct sum of and its orthogonal complement.

Lemma 2.1 (see [21]).

Let be a -quasiclass A operator for a positive integer and let on be matrix expression. Assume that ran is not dense, then is a class A operator on and . Furthermore, .

Proof.

That is, is a class A operator on .

Since , where is the union of the holes in which happen to be subset of by [23, Corollary 7], and and has no interior points, we have .

Theorem 2.2.

Let be a -quasiclass A operator for a positive integer . Then the following assertions hold.

for all and all positive integers .

If for some positive integer , then .

for all positive integers , where denotes the spectral radius of .

To give a proof of Theorem 2.2, the following famous inequality is needful.

Lemma 2.3 (Hölder-McCarthy's inequality [24]).

Let . Then the following assertions hold.

- (1)

Lemma 2.4 (see [21]).

Let be a -quasiclass A operator for a positive integer . If and for some , then .

Proof.

Hence we may write

Hence and . Since is a -quasiclass A operator, by a simple calculation we have

Recall that if and only if and for some contraction . Thus we have . This completes the proof.

Lemma 2.5 (see [25]).

If satisfies for some complex number , then for any positive integer .

Proof.

It suffices to show by induction. We only need to show since is clear. In fact, if , then we have by hypothesis. So we have , that is, . Hence .

An operator is said to have finite ascent if for some positive integer .

Theorem 2.6.

Let be a -quasiclass A operator for a positive integer . Then has finite ascent for all complex number .

Proof.

We only need to show the case because the case holds by Lemmas 2.4 and 2.5.

So we have , which implies . Therefore .

In the following lemma, Tanahashi, Jeon, Kim, and Uchiyama extended the result (1.3) to -quasiclass A operators in the case .

Lemma 2.7 (see [21]).

Let be a -quasiclass A operator for a positive integer . Let be an isolated point of and the Riesz idempotent for . Then the following assertions hold.

An operator is said to be isoloid if every isolated point of is an eigenvalue of .

Theorem 2.8.

Let be a -quasiclass A operator for a positive integer . Then is isoloid.

Proof.

Let be an isolated point. If , by ( ) of Lemma 2.7, for . Therefore is an eigenvalue of . If , by ( ) of Lemma 2.7, for . So we have . Therefore is an eigenvalue of . This completes the proof.

Let denote the tensor product on the product space for nonzero , . The following theorem gives a necessary and sufficient condition for to be a -quasiclass A operator, which is an extension of [20, Theorem 4.2].

Theorem 2.9.

Let , be nonzero operators. Then is a -quasiclass A operator if and only if one of the following assertions holds

(2) and are -quasiclass A operators.

Proof.

Therefore the sufficiency is clear.

To prove the necessary, suppose that is a -quasiclass A operator. Let , be arbitrary. Then we have

From (2.25) we have

where equality holds since is normaloid.

This implies that . Since for all , we have . This contradicts the assumption . Hence must be a -quasiclass A operator. A similar argument shows that is also a -quasiclass A operator. The proof is complete.

## Declarations

### Acknowledgments

The authors would like to express their cordial gratitude to the referee for his useful comments and Professor K. Tanahashi and Professor I. H. Jeon for sending them [21]. This research is supported by the National Natural Science Foundation of China (no. 10771161).

## Authors’ Affiliations

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