Approximate n-idempotents and generalized Aluthge transform


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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  1. 1.

    Aluthge, A.: On \(p\)-hyponormal operators for \(0 < p < 1\). Integral Equ. Oper. Theory 13, 307–315 (1990)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Brown, A.: On a class of operators. Proc. Am. Math. Soc. 4, 723–728 (1953)

    Article  Google Scholar 

  3. 3.

    Chmieliński, J., Ilišević, D., Moslehian, M.S., Sadeghi, G.: Perturbation of the Wigner equation in inner product \(C^*\)-modules. J. Math. Phys. 49(3), 033519 (2008). 8 pp

    MathSciNet  Article  Google Scholar 

  4. 4.

    Furuta, T.: Invitation to Linear Pperators. From Matrices to Bounded Linear Operators on a Hilbert Space. Taylor & Francis Ltd, London (2001)

    Book  Google Scholar 

  5. 5.

    Halmos, P.R.: A Hilbert Space Problem Book. Graduate Texts in Mathematics, 19. Encyclopedia of Mathematics and its Applications, vol. 17, Second edn. Springer, New York (1982)

    Google Scholar 

  6. 6.

    Hyers, D.H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. U.S.A. 27, 222–224 (1941)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Ito, I., Yamazaki, T.: Relations between two inequalities \((B^{\frac{r}{2}}A^pB^{\frac{r}{2}})^{\frac{r}{p+r}}\ge B^r\) and \(A^p\ge (A^{\frac{p}{2}}B^rA^{\frac{p}{2}})^{\frac{p}{p+r}}\) and their applications. Integral Equ. Oper. Theory 44(4), 442–450 (2002)

    Article  Google Scholar 

  8. 8.

    Jarosz, K.: Perturbations of Banach algebras. Lecture Notes in Mathematics, vol. 1120. Springer, Berlin (1985)

    Book  Google Scholar 

  9. 9.

    Jung, I.B., Ko, E., Pearcy, C.: Aluthge transforms of operators. Integral Equ. Oper. Theory 37(4), 437–448 (2000)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Maher, P., Moslehian, M.S.: More on approximate operators. Cubo 14(1), 111–117 (2012)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Mirzavaziri, M., Miura, T., Moslehian, M.S.: Approximate unitaries in \(\cal{B}(H)\). East J. Approx. 16(2), 147–151 (2010)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Miura, T., Uchiyama, A., Oka, H., Hirasawa, G., Takahasi, S.-E., Niwa, N.: A perturbation of normal operators on a Hilbert space. Nonlinear Funct. Anal. Appl. 13(2), 291–297 (2008)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Oloomi, A., Radjabalipour, M.: Operators with normal Aluthge transforms. C. R. Math. Acad. Sci. Paris 350(5–6), 263–266 (2012)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Pedersen, G.K.: Unitary extensions and polar decompositions in a \(C^\ast \)-algebra. J. Oper. Theory 17(2), 357–364 (1987)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Ulam, S.M.: Problems in Modern Mathematics. Chapter VI, Science edn. Wiley, New York (1964)

    MATH  Google Scholar 

  16. 16.

    Zhu, L., Pan, W., Huang, Q., Yang, S.: On the perturbation of outer inverses of linear operators in Banach spaces. Ann. Funct. Anal. 9(3), 344–353 (2018)

    MathSciNet  Article  Google Scholar 

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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.

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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. 94, 979–987 (2020).

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

Mathematics Subject Classification

  • 47A55
  • 39B82
  • 47B15