- Research Article
- Open Access
© 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.
- Positive Integer
- Tensor Product
- Orthonormal Basis
- Space Operator
- Orthogonal Projection
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).
- Stampfli JG: Hyponormal operators and spectral density. Transactions of the American Mathematical Society 1965, 117: 469–476.MATHMathSciNetView ArticleGoogle Scholar
- Chō M, Tanahashi K: Isolated point of spectrum of -hyponormal, log-hyponormal operators. Integral Equations and Operator Theory 2002,43(4):379–384. 10.1007/BF01212700MATHMathSciNetView ArticleGoogle Scholar
- Tanahashi K, Uchiyama A: Isolated point of spectrum of -quasihyponormal operators. Linear Algebra and Its Applications 2002, 341: 345–350. 10.1016/S0024-3795(01)00476-1MATHMathSciNetView ArticleGoogle Scholar
- Tanahashi K, Uchiyama A, Chō M: Isolated points of spectrum of -quasihyponormal operators. Linear Algebra and Its Applications 2004, 382: 221–229.MATHMathSciNetView ArticleGoogle Scholar
- Uchiyama A, Tanahashi K: On the Riesz idempotent of class operators. Mathematical Inequalities & Applications 2002,5(2):291–298.MATHMathSciNetView ArticleGoogle Scholar
- Uchiyama A: On the isolated points of the spectrum of paranormal operators. Integral Equations and Operator Theory 2006,55(1):145–151. 10.1007/s00020-005-1386-0MATHMathSciNetView ArticleGoogle Scholar
- Aluthge A: On -hyponormal operators for . Integral Equations and Operator Theory 1990,13(3):307–315. 10.1007/BF01199886MATHMathSciNetView ArticleGoogle Scholar
- Tanahashi K: On log-hyponormal operators. Integral Equations and Operator Theory 1999,34(3):364–372. 10.1007/BF01300584MATHMathSciNetView ArticleGoogle Scholar
- Arora SC, Arora P: On -quasihyponormal operators for . Yokohama Mathematical Journal 1993,41(1):25–29.MATHMathSciNetGoogle Scholar
- Kim IH: On -quasihyponormal operators. Mathematical Inequalities & Applications 2004,7(4):629–638.MATHMathSciNetView ArticleGoogle Scholar
- Furuta T: On the class of paranormal operators. Proceedings of the Japan Academy 1967, 43: 594–598. 10.3792/pja/1195521514MATHMathSciNetView ArticleGoogle Scholar
- Furuta T: Invitation to Linear Operators: From Matrices to Bounded Linear Operators on a Hilbert Space. Taylor & Francis, London, UK; 2001:x+255.Google Scholar
- Furuta T, Ito M, Yamazaki T: A subclass of paranormal operators including class of log-hyponormal and several related classes. Scientiae Mathematicae 1998,1(3):389–403.MATHMathSciNetGoogle Scholar
- Chō M, Giga M, Huruya T, Yamazaki T: A remark on support of the principal function for class operators. Integral Equations and Operator Theory 2007,57(3):303–308. 10.1007/s00020-006-1463-zMATHMathSciNetView ArticleGoogle Scholar
- Chō M, Yamazaki T: An operator transform from class to the class of hyponormal operators and its application. Integral Equations and Operator Theory 2005,53(4):497–508. 10.1007/s00020-004-1332-6MATHMathSciNetView ArticleGoogle Scholar
- Ito M: Several properties on class including -hyponormal and log-hyponormal operators. Mathematical Inequalities & Applications 1999,2(4):569–578.MATHMathSciNetView ArticleGoogle Scholar
- Ito M, Yamazaki T: Relations between two inequalities and and their applications. Integral Equations and Operator Theory 2002,44(4):442–450. 10.1007/BF01193670MATHMathSciNetView ArticleGoogle Scholar
- Uchiyama A: Weyl's theorem for class operators. Mathematical Inequalities & Applications 2001,4(1):143–150.MATHMathSciNetView ArticleGoogle Scholar
- Wang D, Lee JI: Spectral properties of class operators. Trends in Mathematics Information Center for Mathematical Sciences 2003,6(2):93–98.Google Scholar
- Jeon IH, Kim IH: On operators satisfying . Linear Algebra and Its Applications 2006,418(2–3):854–862. 10.1016/j.laa.2006.02.040MATHMathSciNetView ArticleGoogle Scholar
- Tanahashi K, Jeon IH, Kim IH, Uchiyama A: Quasinilpotent part of class or -quasihyponormal operators. Operator Theory: Advances and Applications 2008, 187: 199–210.MathSciNetGoogle Scholar
- Hansen F: An operator inequality. Mathematische Annalen 1980,246(3):249–250. 10.1007/BF01371046MATHView ArticleGoogle Scholar
- Han JK, Lee HY, Lee WY: Invertible completions of upper triangular operator matrices. Proceedings of the American Mathematical Society 2000,128(1):119–123. 10.1090/S0002-9939-99-04965-5MATHMathSciNetView ArticleGoogle Scholar
- McCarthy CA: “ ” . Israel Journal of Mathematics 1967, 5: 249–271. 10.1007/BF02771613MATHMathSciNetView ArticleGoogle Scholar
- Han YM, Lee JI, Wang D: Riesz idempotent and Weyl's theorem for -hyponormal operators. Integral Equations and Operator Theory 2005,53(1):51–60. 10.1007/s00020-003-1313-1MATHMathSciNetView ArticleGoogle Scholar
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