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Zbl 1230.47042
Cheng, Junxiang; Yuan, Jiangtao
(Cheng, Jun-xiang; Yuan, Jiang-tao)
Aluthge transforms of $(C_{p};\alpha )$-hyponormal operators.
(English)
[J] Ann. Funct. Anal. AFA 2, No. 1, 100-104, electronic only (2011). ISSN 2008-8752/e

Let $\mathcal{H}$ be a separable complex Hilbert space, $B(\mathcal{H})$ be the algebra of all bounded linear operators on $\mathcal{H}$, and for $p\in [1,+\infty)$, let $\mathcal{C}_{p}(\mathcal{H})$ be the ideal of operators in the Schatten $p$-class. For $\alpha>0$, an operator $T$ is said to be $(\mathcal{C}_{p}, \alpha)$-normal if and only if $(T^{*}T)^{\alpha}-(TT^{*})^{\alpha}\in \mathcal{C}_{p}(\mathcal{H})$, and an operator $T$ is said to be $(\mathcal{C}_{p}, \alpha)$-hyponormal if and only if $(T^{*}T)^{\alpha}-(TT^{*})^{\alpha}=P+K$ such that $P$ is a positive semidefinite operator on $\mathcal{H}$ and $K\in \mathcal{C}_{p}(\mathcal{H})$. Let $T\in B(\mathcal{H})$ and $T=U|T|$ be the polar decomposition of $T$. The generalized Aluthge transform $T(s,t)$ is defined by $T(s,t)=|T|^{s}U|T|^{t}$ for $s,t>0$. In this paper, the authors obtain the following result. Let $s>0$, $t>0$, $p\geq 1$ and $\alpha>0$. If $T$ is $(\mathcal{C}_{p}, \alpha)$-hyponormal, then $T(s,t)$ is $(\mathcal{C}_{p(s,t)}, \alpha(s,t))$-hyponormal for $\beta\in (0,1]$, where \par $$p(s,t)=\frac{\max\{2\alpha,s\}p(s+t)}{\min\{\alpha+s,\alpha+t,s+t\}s\beta} \text{ and }\alpha(s,t)=\frac{\min\{\alpha+s,\alpha+t,s+t\}\beta}{s+t}.$$ This result is an extension of [{\it A. Aluthge}, Integral Equations Oper. Theory 13, No.~3, 307--315 (1990; Zbl 0718.47015)], [{\it T. Huruya}, Proc. Am. Math. Soc. 125, No.~12, 3617--3624 (1997; Zbl 0888.47010)], [{\it X.-H. Wang} and {\it Z.-S. Gao}, J. Inequal. Appl. 2010, Article ID 584642 (2010; Zbl 1208.47023)] and [{\it T. Yoshino}, Interdiscip. Inf. Sci. 3, No.~2, 91--93 (1997; Zbl 0928.47013)].
[Takeaki Yamazaki (Kawagoe)]
MSC 2000:
*47B20 Subnormal operators, etc.
47A63 Operator inequalities, etc.

Keywords: Löwner-Heinz inequality; Furuta inequality; hyponormal operators; Aluthge transform; $(\mathcal{C}_{p},\alpha)$-hyponormal; $(\mathcal{C}_{p},\alpha)$-normal; Schatten $p$-class

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