Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

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

Citations: Zbl 0718.47015; Zbl 0888.47010; Zbl 1208.47023; Zbl 0928.47013

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]