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

  • Sid Ahmed Ould Ahmed MahmoudEmail author
  • Muneo Chō
  • Ji Eun Lee


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.


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

Mathematics Subject Classification

Primary 47B20 Secondary 47B99 



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).


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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

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