Powers of Class $p$-$wA(s,t)$ Operators Associated with Generalized Aluthge Transformations

Abstract

Let $T = U|T|$ be a polar decomposition of a bounded linear operator $T$ on a complex Hilbert space with $\ker U = \ker |T|$. $T$ is said to be class $p$-$wA(s,t)$ if
\[
\left( |T^{*}|^{t} |T|^{2s} |T^{*}|^{t} \right)^{\frac{tp}{s+t}} \geq |T^{*}|^{2tp}
\quad\text{and}\quad
|T|^{2sp} \geq \left( |T|^{s} |T^{*}|^{2t} |T|^{s} \right)^{\frac{sp}{s+t}}
\]
with $0 < p \leq 1$ and $0 < s, t, s + t \leq 1$.
This is a generalization of $p$-hyponormal or class $A$ operators. In this paper, we shall show that if $T$ belongs to class $p$-$wA(s,t)$ operator for $0<s,t\leq 1$ and $0<p\leq 1$, then $T^n$ belongs to class $p_1$-$wA\!\left(\frac{s}{n},\frac{t}{n}\right)$ for $0<p_1\leq p$ and for all positive integer $n$.
As an immediate corollary of this result, we shall also show that if $T$ is a $p$-$w$-hyponormal operator, then $T^n$ is also $p_1$-$w$-hyponormal for $0<p_1\leq p$ and for all positive integer $n$.