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Powers of class \(wA(s,t)\) operators associated with generalized Aluthge transformation. (English) Zbl 1050.47018

As a development of operator inequalities, class \(wA(s,t)\) is defined as a class of bounded linear operators on a Hilbert space satisfying \[ (| T^{*}| ^{t}| T| ^{2s}| T^{*}| ^{t})^{\frac{t}{s+t}}\geq | T^{*}| ^{2t} \quad\text{and}\quad | T| ^{2s}\geq (| T| ^{s}| T^{*}| ^{2t}| T| ^{s}) ^{\frac{s}{s+t}} \] for \(s,t>0\) in [M. Ito, SUT J. Math. 35, 149–165 (1999; Zbl 0935.47013)]. This class includes the class of hyponormal operators, and is included in the one of paranormal operators.
In this paper, the author obtains that for \(s, t\in (0,1]\) and \(n\in{\mathbb N}\), if \(T\) belongs to class \(wA(s,t)\), then \(T^{n}\) belongs to the class \(wA(\frac{s}{n}, \frac{t}{n})\) which is a smaller class than class \(wA(s,t)\). This is an extension of the main results in [Y. O. Kim, Nihonkai Math. J. 10, 195–198 (1999; Zbl 1016.47022)] and [M. Chō, T. Huruya and Y. O. Kim, J. Inequal. Appl. 7, 1–10 (2002; Zbl 1044.47017)]. Recently, an extension of the above result was obtained in [M. Ito and T. Yamazaki, Integral Equations Oper. Theory 44, 442–450 (2002; Zbl 1028.47013)].

MSC:

47B20 Subnormal operators, hyponormal operators, etc.
47A63 Linear operator inequalities
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