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The iterated Aluthge Transforms of compact operators

Neeru Bala

TL;DR

We prove that the Aluthge transform $Δ$ is norm-continuous on the space of compact operators $\mathcal{K}(\mathcal{H})$ and that for every $T\in\mathcal{K}(\mathcal{H})$, the iterates $(Δ^n T)$ converge in norm to a normal compact operator $S$ with $\sigma(S)=\sigma(T)$. The convergence is established via collective compactness, polar decomposition arguments, and finite-rank reduction, supplemented by a matrix-case result for finite matrices. This yields an affirmative answer to the Jung–Ko–Pearcy questions in the compact setting and shows that in the quasinilpotent case $Δ^n T\to 0$ in norm. The limiting operator is normal and preserves the spectrum of the original operator.

Abstract

Let $T$ be a bounded linear operator on a Hilbert space. Then the Aluthge transform $ΔT$ and the sequence $(Δ^nT)$ of Aluthge iterates of $T$ are defined by \begin{align*} ΔT=|T|^{1/2}U|T|^{1/2},\,Δ^0T=T,\,Δ^nT=Δ(Δ^{n-1}T),\,n\in\mathbb{N}. \end{align*} We prove that $Δ$ 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 $(Δ^nT)$ 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.

The iterated Aluthge Transforms of compact operators

TL;DR

We prove that the Aluthge transform is norm-continuous on the space of compact operators and that for every , the iterates converge in norm to a normal compact operator with . The convergence is established via collective compactness, polar decomposition arguments, and finite-rank reduction, supplemented by a matrix-case result for finite matrices. This yields an affirmative answer to the Jung–Ko–Pearcy questions in the compact setting and shows that in the quasinilpotent case in norm. The limiting operator is normal and preserves the spectrum of the original operator.

Abstract

Let be a bounded linear operator on a Hilbert space. Then the Aluthge transform and the sequence of Aluthge iterates of are defined by \begin{align*} ΔT=|T|^{1/2}U|T|^{1/2},\,Δ^0T=T,\,Δ^nT=Δ(Δ^{n-1}T),\,n\in\mathbb{N}. \end{align*} We prove that 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 converges in the norm topology to a normal compact operator for every compact operator on a separable Hilbert space. This gives an affirmative answer to two questions raised by Jung, Ko and Pearcy \cite{Pearcy2} for compact operators.
Paper Structure (2 sections, 9 theorems, 12 equations)

This paper contains 2 sections, 9 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Iterated Aluthge transform of compact operators

Key Result

Theorem 1.1

For every $T\in M_r(\mathbb{C})$, the sequence $(\Delta^nT)$ converges to a normal matrix, where $M_r(\mathbb{C})$ is the algebra of all complex $r\times r$ matrices. Moreover, the spectrum of limiting matrix is same as the spectum of $T$.

Theorems & Definitions (14)

  • Theorem 1.1
  • Theorem 1.4
  • Proposition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Remark 2.3
  • ...and 4 more