Properties of two unitary operator functions involving idempotents

  • Jiaxin Zhang
  • Jiajia Niu
  • Yuan LiEmail author
Original Paper


Let P be an idempotent operator on a Hilbert space \(\mathcal {H}.\) We denote two unitary operator functions \(U_{\lambda }\) and \(V_{\lambda }\) by
$$\begin{aligned} U_{\lambda }:=(\lambda P+I)|\lambda P+I|^{-1} \hbox { } \hbox { and }\hbox { } V_{\lambda }:=(\lambda P^{*}+I)|\lambda P^{*}+I|^{-1}, \ \ \hbox { for }\lambda \in \mathbb {C}\backslash \{-1\}. \end{aligned}$$
In this paper, we first give the specific structures of \(U_{\lambda }\) and \(V_{\lambda },\) respectively. Then the sufficient and necessary conditions under which \(U_{\lambda }\) and \(V_{\lambda }\) are symmetries are presented. Moreover, the specific structures and spectra of the unitary operator \(U=\lim \limits _{\lambda \rightarrow -1^+}U_{\lambda }\) are characterized.


Idempotents Symmetries The operator function 

Mathematics Subject Classification

47A05 47A62 46C20 



The authors would like to express their heart-felt thanks to the anonymous referees for some valuable comments. This work was supported by NSF of China (Nos: 11671242, 11571211) and the Fundamental Research Funds for the Central Universities (GK201801011)


  1. 1.
    Andruchow, E.: Classes of idempotent in Hilbert Space. Complex Anal. Oper. Theory 10, 1383–1409 (2016)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Azizov, T.Ya., Iokhvidov, I.S.: Linear Operators in Spaces with an Indefinite Metric, vol. 304. Wiley, Chichester (1989)zbMATHGoogle Scholar
  3. 3.
    Ando, T.: Projections in Krein spaces. Linear Algebra Appl. 12, 2346–2358 (2009)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Arias, M.L., Corach, G., Maestripieri, A.: Products of idempotent operators. Integr. Equ. Oper. Theory 88, 269–286 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Buckholtz, D.: Hilbert space idempotents and involution. Proc. Am. Math. 128, 1415–1418 (1999)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Böttcher, A., Simon, B., Spitkovsky, I.: Similarity between two projections. Integr. Equ. Oper. Theory 89, 507–518 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Corach, G., Porta, H., Recht, L.: The geometry of spaces of projections in C*-algebras. Adv. Math. 101, 59–77 (1993)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Corach, G., Maestripieri, A., Stojanoff, D.: Oblique projections and Schur complements. Acta Sci. Math. (Szeged) 67, 337–356 (2001)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Li, Y., Zhang, J. X., Wei, N. N.: The structures and decompositions of symmetries involving idempotents. arXiv:1903.01746
  10. 10.
    Li, Y., Cai, X.M., Wang, S.J.: The absolute values and support projections for a class of operator matrices involving idempotents. Complex Anal. Oper. Theory 13, 1949–1973 (2019)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Li, Y., Cai, X.M., Niu, J.J., Zhang, J.X.: The minimal and maximal symmetries for \(J\)-contractive projections. Linear Algebra Appl. 563, 313–330 (2019)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Maestripieri, A., Pería, F.M.: Normal projections in Krein Spaces. Integr. Equ. Oper. Theory 76, 357–380 (2013)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Matvejchuk, M.: Idempotents and Krein space. Lobachevskii J. Math. 32(2), 128–134 (2011)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Matvejchuk, M.: Idempotents as \(J\)-Projections. Int. J. Theor Phys. 50, 3852–3856 (2011)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Matvejchuk, M.: Idempotents in a space with conjugation. Linear Algebra Appl. 438, 71–79 (2013)MathSciNetCrossRefGoogle Scholar

Copyright information

© Tusi Mathematical Research Group (TMRG) 2019

Authors and Affiliations

  1. 1.School of Mathematics and Information ScienceShaanxi Normal UniversityXi’anPeople’s Republic of China

Personalised recommendations