© F. Gao and X. Fang. 2009
Received: 26 June 2009
Accepted: 10 November 2009
Published: 13 December 2009
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.
After that many authors extended this result to many other classes of operators. Chō and Tanahashi  proved that (1.3) holds if is either -hyponormal or log-hyponormal. In the case , the result was further shown by Tanahashi and Uchiyama  to hold for -quasihyponormal operators, by Tanahashi et al.  to hold for -quasihyponormal operators and by Uchiyama and Tanahashi  and Uchiyama  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 , Tanahashi , S. C. Arora and P. Arora , Kim , and Furuta [11, 12], respectively.
In order to discuss the relations between paranormal and -hyponormal and log-hyponormal operators, Furuta et al.  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  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 , 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 .
In , this class of operators is called quasi-class (A, ).
We show that the inclusion relation (1.5) is strict, by an example which appeared in .
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.
Lemma 2.1 (see ).
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 .
To give a proof of Theorem 2.2, the following famous inequality is needful.
Lemma 2.3 (Hölder-McCarthy's inequality ).
Lemma 2.4 (see ).
Hence we may write
Lemma 2.5 (see ).
Lemma 2.7 (see ).
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].
Therefore the sufficiency is clear.
From (2.25) we have
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.
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 . This research is supported by the National Natural Science Foundation of China (no. 10771161).
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