Advertisement

On \(\varvec{(m,C)}\)-Isometric Commuting Tuples of Operators on a Hilbert Space

  • Sid Ahmed Ould Ahmed MahmoudEmail author
  • Muneo Chō
  • Ji Eun Lee
Article
  • 65 Downloads

Abstract

Inspired by recent works on (mC)-isometric and [mC]-isometric operators on Hilbert spaces studied respectively in Chō et al. (Complex Anal. Oper. Theory 10:1679–1694, 2016; Filomat 31:7, 2017), in this paper we introduce the class of (mC)-isometries for tuple of commuting operators. This is a generalization of the class of (mC)-isometric operators. A commuting tuples of operators \(\mathbf{\large T}=(T_1,\ldots ,T_d)\in {\mathcal {B}}^{(d)}({\mathcal {H}})\) is said to be (mC)-isometric tuple if
$$\begin{aligned} {{\mathcal {Q}}}_{m}(\mathbf{T}):=\sum _{0\le k\le m}(-1)^{m-k}\left( {\begin{array}{c}m\\ k\end{array}}\right) \left( \sum _{|\beta |=k}\frac{k!}{\beta !}\mathbf{\large T}^{*\beta }C\mathbf{\large T}^{\beta }C\right) =0 \end{aligned}$$
for some positive integer m and some conjugation C. We consider a multivariable generalization of these single variable (mC)-isometric operators and explore some of their basic properties.

Keywords

\((m, C)\)-isometric operator \((m, C)\)-isometric tuple of Hilbert space 

Mathematics Subject Classification

Primary 47B20 Secondary 47B99 

Notes

Acknowledgements

This research is partially supported by Grant-in-Aid Scientific Research No.15K04910. The third author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (2016R1A2B4007035).

References

  1. 1.
    Agler, J., Stankus, M.: \(m\)-Isometric transformations of Hilbert space I. Integral Equ. Oper. Theory 21, 383–429 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ambrozie, C., Englis, M., Müller, V.: Operator tuples and analytic models over general domains in \({\mathbb{C}}^n\). J. Oper. Theory 47, 287–302 (2002)zbMATHGoogle Scholar
  3. 3.
    Ben Amor, A.: An extension of Henrici theorem for the joint approximate spectrum of commuting spectral operators. J. Aust. Math. Soc. 75, 233–245 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bermúdez, T., Martinón, A., Müller, V.: \((m, q)\)-isometries on metric spaces. J. Oper. Theory 72(2), 313–329 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chō, M., Curto, R.E., Huruya, T.: \(n\)-Tuples of operators satisfying \(\sigma _T (AB) = \sigma _T(BA)\). Linear Algebra Appl. 341, 291–298 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chō, M., Jeon, I.H., Jung, I.B., Lee, J.I., Tanahashi, K.: Joint spectra of \(n\)-tuples of generalized Aluthge transformations. Rev. Roumaine Math. Pures Appl. 46(6), 725–730 (2001)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Chō, M., Jeon, I.H., Lee, J.I.: Joint spectra of doubly commuting \(n\)-tuples of operators and their Aluthge transforms. Nihonkai Math. J. 11(1), 87–96 (2000)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Chō, M., Ko, E., Lee, J.E.: On \((m, C)\)-isometric operators. Complex Anal. Oper. Theory 10, 1679–1694 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chō, M., Ko, E., Lee, J.E.: \((\infty, C)\)-isometric operators. Oper. Matrices 113, 793–806 (2017)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Chō, M., Lee, J.E., Motoyoshi, H.: On \([m, C]\)-isometric operators. Filomat 31(7), 2073–2080 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Chō, M., Müller, V.: Spectral commutativity of multioperators. Funct. Anal. Approx. Comput. 4(1), 21–25 (2012)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Chō, M., Żelazko, W.: On geometric spectral radius of commuting \(n\)-tuples of operators. Hokkaido Math. J. 21(2), 251–258 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gleason, J., Richter, S.: \(m\)-Isometric commuting tuples of operators on a Hilbert space. Integral Equ. Oper. Theory 56(2), 181–196 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gu, C.: The \((m, q)\)-isometric weighted shifts on \(l_p\) spaces. Integral Equ. Oper. Theory 82, 157–187 (2015)CrossRefzbMATHGoogle Scholar
  15. 15.
    Hoffmann, P.H.W., Mackey, M.: \((m, p)\)-and \((m,\infty )\)-isometric operator tuples on normed spaces. Asian Eur. J. Math. 8(2), 1–32 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Sid Ahmed Ould, A. M., Chō, M, Lee, J. E.: On \(n\)-quasi-\((m,C)\)-isometric operators. (Review)Google Scholar
  17. 17.
    Slodkowski, Z., Żelazko, W.: On joint spectra of commuting families of operators. Studia Math. 50, 127–148 (1974)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Sid Ahmed Ould Ahmed Mahmoud
    • 1
    Email author
  • Muneo Chō
    • 2
  • Ji Eun Lee
    • 3
  1. 1.Mathematics Department, College of ScienceJouf UniversityAljoufSaudi Arabia
  2. 2.Department of MathematicsKanagawa UniversityHiratsukaJapan
  3. 3.Department of Mathematics and StatisticsSejong UniversitySeoulKorea

Personalised recommendations