Open Access

On -Quasiclass A Operators

Journal of Inequalities and Applications20092009:921634

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

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.

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

(1.1)

Let

(1.2)
is called the Riesz idempotent with respect to , and it is well known that satisfies , , , and for all positive integers . Stampfli [1] proved that if is hyponormal (i.e., operators such that ), then
(1.3)

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, 1419].

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.

is called a -quasiclass A operator for a positive integer if
(1.4)

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

(1.5)

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

Example 1.3.

Given a bounded sequence of positive numbers , let be the unilateral weighted shift operator on with the canonical orthonormal basis by for all , that is,
(1.6)

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.

Consider the matrix representation of with respect to the decomposition : Let be the orthogonal projection of onto . Then . Since is a -quasiclass A operator, we have
(2.1)
Then
(2.2)
by Hansen's inequality [22]. On the other hand
(2.3)
Hence
(2.4)

That is, is a class A operator on .

For any ,

(2.5)

which implies .

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.

for and all .

for and all .

Proof of Theorem 2.2.
  1. (1)
    Since it is clear that -quasiclass A operators are -quasiclass A operators, we only need to prove the case . Since
    (2.6)
     
by Hölder-McCarthy's inequality, we have
(2.7)
for is a -quasiclass A operator.
  1. (2)

    If , it is obvious that . If , then by ( ). The rest of the proof is similar.

     
  2. (3)

    We only need to prove the case , that is,

     
(2.8)
If for some , then by (2) and in this case . Hence (3) is clear. Therefore we may assume for all . Then
(2.9)
by ( ), and we have
(2.10)
Hence
(2.11)
By letting , we have
(2.12)
that is,
(2.13)

Lemma 2.4 (see [21]).

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

Proof.

We may assume that . Let be a span of . Then is an invariant subspace of and
(2.14)
Let be the orthogonal projection of onto . It suffices to show that in (2.14). Since is a -quasiclass A operator, and , we have
(2.15)
We remark
(2.16)
Then by Hansen's inequality and (2.15), we have
(2.17)

Hence we may write

(2.18)
We have
(2.19)
This implies and . On the other hand,
(2.20)

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

(2.21)

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.

In the case , we shall show that . It suffices to show that since is clear. Now assume that . We may assume since if , it is obvious that . By Hölder-McCarthy's inequality, we have
(2.22)

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.

If , then is self-adjoint and

(2.23)

If , then .

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

(1) or .

(2) and are -quasiclass A operators.

Proof.

It is clear that is a -quasiclass A operator if and only if
(2.24)

Therefore the sufficiency is clear.

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

(2.25)
It suffices to prove that if ( ) does not hold, then ( ) holds. Suppose that and . To the contrary, assume that is not a -quasiclass A operator, then there exists such that
(2.26)

From (2.25) we have

(2.27)
that is,
(2.28)
for all . Therefore is a -quasiclass A operator. As the proof in Theorem 2.2 ( ), we have
(2.29)
So we have
(2.30)
for all by (2.28). Because is a -quasiclass A operator, from Lemma 2.1 we can write on , where is a class A operator (hence it is normaloid). By (2.30) we have
(2.31)
So we have
(2.32)

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

(1)
Department of Mathematics, Tongji University
(2)
College of Mathematics and Information Science, Henan Normal University

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Copyright

© F. Gao and X. Fang. 2009

This article is published under license to BioMed Central Ltd. 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.