## Annals of Functional Analysis

### Aluthge Transforms of $(\mathcal{C}_{p},\alpha)$-Hyponormal Operators

#### Abstract

Recently, the class of $(\mathcal{C}_{p},\alpha)$-hyponormal operators is introduced and the Aluthge transforms of such operators is discussed by some researchers. This paper is to give a further development of the Aluthge transforms of $(\mathcal{C}_{p},\alpha)$-hyponormal operators by using Loewner-Heinz inequality, Furuta inequality and Lauric's lemma. Especially, it is shown that, if $p\ge 1$, $\alpha\ge 1/2$ and $T$ is $(\mathcal{C}_{p},\alpha)$-hyponormal, then the Aluthge transform $T(1/2,1/2)$ is $(\mathcal{C}_{4p\alpha/\beta},\beta)-hyponormal$ where $0 \lt \beta \le 1$ and $T(1/2,1/2)=|T|^{1/2}U|T|^{1/2}$.

#### Article information

Source
Ann. Funct. Anal., Volume 2, Number 1 (2011), 100-104.

Dates
First available in Project Euclid: 12 May 2014

https://projecteuclid.org/euclid.afa/1399900265

Digital Object Identifier
doi:10.15352/afa/1399900265

Mathematical Reviews number (MathSciNet)
MR2811210

Zentralblatt MATH identifier
1230.47042

Subjects
Primary: 47B20: Subnormal operators, hyponormal operators, etc.
Secondary: 47A63: Operator inequalities

#### Citation

Cheng, Junxiang; Yuan, Jiangtao. Aluthge Transforms of $(\mathcal{C}_{p},\alpha)$-Hyponormal Operators. Ann. Funct. Anal. 2 (2011), no. 1, 100--104. doi:10.15352/afa/1399900265. https://projecteuclid.org/euclid.afa/1399900265

#### References

• A. Aluthge, On $p$–hyponormal operators for $0<p<1$, Integral Equations Operator Theory 13 (1990), 307–315.
• M. Fujii, Furuta inequality and its related topics, Ann. Funct. Anal. 1 (2010), 28–45.
• T. Furuta, Invitation to Linear Operators, Taylor & Francis, London, 2001.
• T. Huruya, A note on $p$–hyponormal operators, Proc. Amer. Math. Soc., 125 (1997), 3617–3624.
• V. Lauric, $(C_{p}, \alpha)$–hyponormal operators and trace–class self–commutators with trace zero, Proc. Amer. Math. Soc. 137 (2009), 945–953.
• X. Wang and Z. Gao, A note on $(C_{p},\alpha)$–hyponormal operators, J. Inequal. Appl. 2010 (2010), Article ID 584642, 10 pages.
• M. Yanagida, Some applications of Tanahashi's result on the best possibility of Furuta inequality, Math. Inequal. Appl. 2 (1999), 297–305.
• T. Yoshino, The $p$–hyponormality of the Aluthge transform, Interdiscip. Inform. Sci. 3 (1997), 91–93.
• J. Yuan, Furuta inequality and $q$–hyponormal operators, Oper. Matrices 4 (3) (2010), 405–415.
• J. Yuan and Z. Gao, Complete form of Furuta inequality, Proc. Amer. Math. Soc. 136 (8) (2008), 2859–2867.