The iterated Aluthge Transforms of compact operators

TL;DR

This paper proves that the iterated Aluthge transform of a compact operator on a separable Hilbert space converges in norm to a normal compact operator, confirming previous open questions.

Contribution

It establishes the continuity of the Aluthge transform on compact operators and proves convergence of its iterates in the norm topology.

Findings

The Aluthge transform is continuous on the space of compact operators.

Iterates of the Aluthge transform converge in norm to a normal compact operator.

Answers two open questions for compact operators regarding convergence.

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.