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On Quasiclass A Operators
Journal of Inequalities and Applications volume 2009, Article number: 921634 (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
Let
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
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 loghyponormal. 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 loghyponormal 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, loghyponormal, 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 loghyponormal 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 loghyponormal 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 quasiclass 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
Remark 1.2.
In [21], this class of operators is called quasiclass (A, ).
It is clear that the class of quasiclassA operatorsthe class of kquasiclass A operators and
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,
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 nonzero 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
Then
by Hansen's inequality [22]. On the other hand
Hence
That is, is a class A operator on .
For any ,
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 (HölderMcCarthy's inequality [24]).
Let . Then the following assertions hold.
for and all .
for and all .
Proof of Theorem 2.2.

(1)
Since it is clear that quasiclass A operators are quasiclass A operators, we only need to prove the case . Since
(2.6)
by HölderMcCarthy's inequality, we have
for is a quasiclass A operator.

(2)
If , it is obvious that . If , then by (). The rest of the proof is similar.

(3)
We only need to prove the case , that is,
If for some , then by (2) and in this case . Hence (3) is clear. Therefore we may assume for all . Then
by (), and we have
Hence
By letting , we have
that is,
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
Let be the orthogonal projection of onto . It suffices to show that in (2.14). Since is a quasiclass A operator, and , we have
We remark
Then by Hansen's inequality and (2.15), we have
Hence we may write
We have
This implies and . On the other hand,
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.
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 HölderMcCarthy's inequality, we have
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 selfadjoint and
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
Therefore the sufficiency is clear.
To prove the necessary, suppose that is a quasiclass A operator. Let , be arbitrary. Then we have
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
From (2.25) we have
that is,
for all . Therefore is a quasiclass A operator. As the proof in Theorem 2.2 (), we have
So we have
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
So 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.
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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).
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Gao, F., Fang, X. On Quasiclass A Operators. J Inequal Appl 2009, 921634 (2009). https://doi.org/10.1155/2009/921634
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DOI: https://doi.org/10.1155/2009/921634