The iterated Aluthge Transforms of compact operators

Abstract

Let $T$ be a bounded linear operator on a Hilbert space. Then the Aluthge transform $\Delta T$ and the sequence $(\Delta^nT)$ of Aluthge iterates of $T$ are defined by \begin{align*} \Delta T=|T|^{1/2}U|T|^{1/2},\,\Delta^0T=T,\,\Delta^nT=\Delta(\Delta^{n-1}T),\,n\in\mathbb{N}. \end{align*} We prove that $\Delta$ is a continuous map on the space of all compact operators on a separable Hilbert space with respect to the norm topology and using this result we also prove that the sequence $(\Delta^nT)$ converges in the norm topology to a normal compact operator for every compact operator $T$ on a separable Hilbert space. This gives an affirmative answer to two questions raised by Jung, Ko and Pearcy \cite{Pearcy2} for compact operators.