The $\lambda$- Aluthge transform of EP matrices

Abstract

In this work we present some relationships between an EP matrix $T$, its Aluthge transform $\Delta(T)$ or the $\lambda$-Aluthge transform $\Delta_{\lambda}(T)$   and the Moore-Penrose inverse $T^\dagger$. We prove that the $\lambda$-Aluthge transform of $T$ is also an EP matrix, and the same thing holds for $\Delta_{\lambda}(T)^\dagger$ and $\Delta_{\lambda}(T^\dagger)$. Also, we explore the product $\Delta_{\lambda}(T) T$, the connections between $\Delta(T)$ and $T^\dagger$ as well as the reverse order law for generalized inverses which are associated with $\Delta_{\lambda}(T)$. Finally, it is verified that the ranges of $T$ and $\Delta_{\lambda}(T)$ are equal in the case of EP matrices.