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Approximate n-idempotents and generalized Aluthge transform

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Abstract

Let p be a real number and let \(\varepsilon >0\). An operator \(T\in \mathbb {B}(\mathscr {H})\) is called a \((p,\varepsilon )\)-approximate n-idempotent if

$$\begin{aligned} \Vert T^nx- Tx\Vert \le \varepsilon \Vert x\Vert ^p\qquad (x\in \mathscr {H})\,. \end{aligned}$$

In this note, we remark that if \(p\ne 1\), then T is an n-idempotent. If \(p=1\), the operator T is a self-adjoint contraction satisfying \((-T)^n\ge 0\), and \(\varepsilon < \frac{n-1}{n\,\root n-1 \of {n}}\), then there is a self-adjoint n-idempotent S such that \(\Vert T-S\Vert < K\varepsilon \) for some constant \(K>0\). Among other results, we examine the lack of a similar result for the \((1,\varepsilon )\)-approximate generalized Aluthge transform.

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Acknowledgements

The author would like to sincerely thank Professor Takeaki Yamazaki and the referee for their useful comments. This research is supported by a Grant from Ferdowsi University of Mashhad (No. 2/51506).

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Correspondence to Mohammad Sal Moslehian.

Additional information

Dedicated to Professor Asadollah Niknam on his 70th birthday with respect and affection.

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Moslehian, M.S. Approximate n-idempotents and generalized Aluthge transform. Aequat. Math. (2020). https://doi.org/10.1007/s00010-020-00713-6

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Keywords

  • Generalized Aluthge transform
  • n-idempotent
  • Quasinormal operator
  • Stability

Mathematics Subject Classification

  • 47A55
  • 39B82
  • 47B15