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

Abstract

Let $T = U \vert T \vert $ be a polar decomposition of a bounded linear operator $T$ on a complex Hilbert space with $\ker U = \ker \vert T \vert$. $T$ is said to be class $p$-$wA(s,t)$ if
$ \left( \vert T^{*} \vert^{t} \vert T \vert^{2s} \vert T^{*} \vert^{t} \right)^{\frac{tp}{s+t}}
 \geq \vert T^{*} \vert^{2tp}$ and $ \vert T \vert^{2sp} \geq \left( \vert T \vert^{s} \vert T^{*} \vert^{2t} \vert T \vert^{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 $p_1$-$wA(\frac{s}{n},\frac{t}{n})$ 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 $n$.