Israel Journal of Mathematics

, Volume 223, Issue 1, pp 343–362 | Cite as

Angle criteria for uniform convergence of averaged projections and cyclic or random products of projections

  • Izhar OppenheimEmail author


We apply a new notion of angle between projections to deduce criteria for uniform convergence results of the alternating projections method under several different settings: averaged projections, cyclic products, quasiperiodic products and random products.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    I. Amemiya and T. Andô, Convergence of random products of contractions in Hilbert space, Acta Sci. Math. (Szeged) 1965 (1965), 239᾿44.MathSciNetzbMATHGoogle Scholar
  2. [2]
    C. Badea, S. Grivaux and V. Müller, The rate of convergence in the method of alternating projections, Algebra i Analiz 2011 (2011), 1᾿0.zbMATHGoogle Scholar
  3. [3]
    C. Badea and Y. I. Lyubich, Geometric, spectral and asymptotic properties of averaged products of projections in Banach spaces, Studia Math. 2010 (2010), 21᾿5.zbMATHGoogle Scholar
  4. [4]
    J. Dye, M. A. Khamsi and S. Reich, Random products of contractions in Banach spaces, Trans. Amer. Math. Soc. 1991 (1991), 87᾿9.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    J. Dye, Convergence of random products of compact contractions in Hilbert space, Integral Equations Operator Theory 1989 (1989), 12᾿2.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    J. M. Dye, T. Kuczumow, P.-K. Lin and S. Reich, Convergence of unrestricted products of nonexpansive mappings in spaces with the Opial property, Nonlinear Anal. 1996 (1996), 767᾿73.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    J. M. Dye and S. Reich, On the unrestricted iteration of projections in Hilbert space, J. Math. Anal. Appl. 1991 (1991), 101᾿19.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    J. M. Dye and S. Reich, Unrestricted iterations of nonexpansive mappings in Banach spaces, Nonlinear Anal. 1992 (1992), 983᾿92.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    I. Halperin, The product of projection operators, Acta Sci. Math. (Szeged) 1962 (1962), 96᾿9.MathSciNetzbMATHGoogle Scholar
  10. [10]
    E. Kopecká and V. Müller, A product of three projections, Studia Math. 2014 (2014), 175᾿86.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    E. Kopecká and A. Paszkiewicz, Strange products of projections, Israel J. Math. 2017 (2017), 271᾿86.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    M. L. Lapidus, Generalization of the Trotter-Lie formula, Integral Equations Operator Theory 1981 (1981), 366᾿15.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    I. Oppenheim, Vanishing of cohomology with coefficients in representations on Banach spaces of groups acting on buildings, Commentarii Mathematici Helvetici, to appear.Google Scholar
  14. [14]
    I. Oppenheim, Averaged projections, angles between groups and strengthening of Banach property (T), Math. Ann. 2017 (2017), 623᾿66.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    M. I. Ostrovskiĭ, Topologies on the set of all subspaces of a Banach space and related questions of Banach space geometry, Quaestiones Math. 1994 (1994), 259᾿19.MathSciNetCrossRefGoogle Scholar
  16. [16]
    A. Paszkiewicz, The amemiya-ando conjecture falls, 3354v1.pdf, 2012.Google Scholar
  17. [17]
    E. Pustylnik, S. Reich and A. J. Zaslavski, Convergence of non-cyclic infinite products of operators, J. Math. Anal. Appl. 2011 (2011), 759᾿67.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    E. Pustylnik, S. Reich and A. J. Zaslavski, Convergence of non-periodic infinite products of orthogonal projections and nonexpansive operators in Hilbert space, J. Approx. Theory 2012 (2012), 611᾿24.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    E. Pustylnik, S. Reich and A. J. Zaslavski, Inner inclination of subspaces and infinite products of orthogonal projections, J. Nonlinear Convex Anal. 2013 (2013), 423᾿36.MathSciNetzbMATHGoogle Scholar
  20. [20]
    S. Reich, A limit theorem for projections, Linear and Multilinear Algebra 1983 (1983), 281᾿90.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    M. Sakai, Strong convergence of infinite products of orthogonal projections in Hilbert space, Appl. Anal. 1995 (1995), 109᾿20.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    J. von Neumann, On rings of operators. Reduction theory, Ann. of Math. (2) 1949 (1949), 401᾿85.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 2018

Authors and Affiliations

  1. 1.Department of MathematicsBen-Gurion University of the NegevBe’er ShevaIsrael

Personalised recommendations