Israel Journal of Mathematics

, Volume 219, Issue 1, pp 271–286 | Cite as

Strange products of projections



Let H be an infinite-dimensional Hilbert space. We show that there exist three orthogonal projections X 1,X 2,X 3 onto closed subspaces of H such that for every 0z 0H there exist k 1, k 2, · · · ∈ {1, 2, 3} so that the sequence of iterates defined by z n = X kn z n −1 does not converge in norm.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    I. Amemiya and T. Ando, Convergence of random products of contractions in Hilbert space, Acta Sci. Math. (Szeged) 26 (1965), 239–244.MathSciNetMATHGoogle Scholar
  2. [2]
    I. Halperin, The product of projection operators, Acta Sci. Math. (Szeged) 23 (1962), 96–99.MathSciNetMATHGoogle Scholar
  3. [3]
    H. S. Hundal, An alternating projection that does not converge in norm, Nonlinear Anal. 57 (2004), 35–61.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    E. Kopecká, Spokes, mirrors and alternating projections, Nonlinear Anal. 68 (2008), 1759–1764.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    E. Kopecká and V. Müller, A product of three projections, Studia Math. 223 (2014), 175–186.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    E. Kopecká and S. Reich, A note on the von Neumann alternating projections algorithm, J. Nonlinear Convex Anal. 5 (2004), 379–386.MathSciNetMATHGoogle Scholar
  7. [7]
    E. Matoušková and S. Reich, The Hundal example revisited, J. Nonlinear Convex Anal. 4 (2003), 411–427.MathSciNetMATHGoogle Scholar
  8. [8]
    A. Paszkiewicz, The Amemiya–Ando conjecture falls, arXiv:1203.3354.Google Scholar
  9. [9]
    M. Práger, Über ein Konvergenzprinzip im Hilbertschen Raum, Czechoslovak Math. J. 10 (1960), 271–282, in Russian.MathSciNetMATHGoogle Scholar
  10. [10]
    M. Sakai, Strong convergence of infinite products of orthogonal projections in Hilbert space, Appl. Anal. 59 (1995), 109–120.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    J. von Neumann, On rings of operators. Reduction theory, Ann. of Math. 50 (1949), 401–485.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of InnsbruckInnsbruckAustria
  2. 2.Faculty of Mathematics and Computer ScienceŁódź UniversityŁódźPoland

Personalised recommendations