The iterated Aluthge Transforms of compact operators
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866911642275545088 |
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| author | Bala, Neeru |
| author_facet | Bala, Neeru |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_07916 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The iterated Aluthge Transforms of compact operators Bala, Neeru Functional Analysis 47B02, 47B07 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. |
| title | The iterated Aluthge Transforms of compact operators |
| topic | Functional Analysis 47B02, 47B07 |
| url | https://arxiv.org/abs/2602.07916 |