The $\lambda$-Aluthge transform and its applications to some classes of operators

Abstract

Let $ T \in \mathcal{B}(\mathcal{H})$ be a bounded linear operator on a Hilbert space $\mathcal{H}$,and $ T = U \vert T \vert$ be its polar decomposition. We say that $ T $ belongs to the class $ \delta ( \mathcal{H}) $ if $ U^{2} \vert T \vert = \vert T \vert U^{2} $. For $\lambda \in [0, 1]$, the $\lambda$-Aluthge transform of $ T $ is defined by $ \Delta_{\lambda}(T) = \vert T \vert ^{\lambda}U\vert T \vert^{1-\lambda}$. It is well known that $ T $ is invertible if and only if $ \Delta_{\lambda}(T) $ is invertible. In general $ \Delta_{\lambda}(T^{-1})\neq (\Delta_{\lambda}(T))^{-1} $. In this paper, we show that an invertible operator $ T $ belongs to $ \delta ( \mathcal{H}) $ if and only if $ \Delta_{1}(T^{-1}) = (\Delta_{1}(T))^{- 1 } $. Moreover, for $\lambda \in ]0, 1[$, we prove that if $ T $ is invertible in $ \delta ( \mathcal{H}) $, then $ T $ is binormal if and only if $ \Delta_{\lambda}(T^{-1})= (\Delta_{\lambda}(T))^{-1} $. Also in this paper, We discuss the $\lambda$-Aluthge transform of EP and binormal operators via Moore-Penrose inverse. In particular, we show that $ T $ is EP if and only if $\Delta_{\lambda}(T) $ is EP too and $ R(T)=R( \Delta_{\lambda}(T) ). $