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

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 $R(T)=R( \Delta_{\lambda}(T) ).$