Paul Halmos and Invariant Subspaces

  • Heydar Radjavi
  • Peter Rosenthal
Part of the Operator Theory Advances and Applications book series (OT, volume 207)


This paper consists of a discussion of the contributions that Paul Halmos made to the study of invariant subspaces of bounded linear operators on Hilbert space.

Mathematics Subject Classification (2000)



Invariant subspaces bounded linear operators 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    C. Apóstol, C. Foias, and D. Voiculescu, Some results on non-quasitriangular operators TV, Rev. Roumaine Math. Pures Appl. 18 (1973), 487–514.zbMATHGoogle Scholar
  2. [2]
    N. Aronszajn and K.T. Smith, Invariant subspaces of completely continuous operators, Ann. of Math. 60 (1954), 345–350.CrossRefMathSciNetGoogle Scholar
  3. [3]
    W.B. Arveson, A density theorem for operator algebras, Duke Math. J. 34 (1967), 635–647.CrossRefzbMATHMathSciNetGoogle Scholar
  4. [4]
    W.B. Arveson, Operator algebras and invariant subspaces, Ann. of Math. (2) 100 (1974), 433–532.Google Scholar
  5. [5]
    W.B. Arveson Ten lectures on operator algebras, CBMS Regional Conference Series in Mathematics 55, A.M.S., Providence, 1984.Google Scholar
  6. [6]
    W.B. Arveson and J. Feldman, A note on invariant subspaces, Michigan Math. J. 15 (1968), 60–64.MathSciNetGoogle Scholar
  7. [7]
    J. Barría, The invariant subspaces of a Volterra operator, J. Op. Th. 6 (1981), 341–349.zbMATHGoogle Scholar
  8. [8]
    J. Barría and K.R. Davidson, Unicellular operators, Trans. Amer. Math. Soc. 284 (1984), 229–246.CrossRefzbMATHMathSciNetGoogle Scholar
  9. [9]
    Charles A. Berger, Sufficiently high powers of hyponormal operators have rationally invariant subspaces, Int. Equat. Op. Th. 1 (1978), 444–447.CrossRefzbMATHGoogle Scholar
  10. [10]
    A.R. Bernstein and A. Robinson, Solution of an invariant subspace problem of K.T. Smith and P.R. Halmos, Pacific J. Math, 16 (1966), 421–431.zbMATHMathSciNetGoogle Scholar
  11. [11]
    A. Beurling, On two problems concerning linear transformations in Hilbert space, Acta Math. 81 (1949), 239–255.CrossRefzbMATHGoogle Scholar
  12. [12]
    A.S. Brodskii, On a problem of I.M. Gelfand, Uspehi Mat. Nauk., 12 (1957), 129–132.MathSciNetGoogle Scholar
  13. [13]
    Arlen Brown and P.R. Halmos, Algebraic properties of Toeplitz operators, J. Reine Angew. Math. 231 (1963), 89–102.MathSciNetGoogle Scholar
  14. [14]
    Scott W. Brown, Some invariant subspaces for subnormal operators, Int. Equat. Oper. Th. 1 (1978), 310–333.CrossRefzbMATHGoogle Scholar
  15. [15]
    Scott W. Brown, Hyponormal operators with thick spectra have invariant subspaces, Ann. of Math. (2) 125 (1987), 93–103.Google Scholar
  16. [16]
    Scott W. Brown, Bernard Chevreau, and Carl Pearcy, Contractions with rich spectrum have invariant subspaces, J. Op. Th. 1 (1979), 123–136.zbMATHMathSciNetGoogle Scholar
  17. [17]
    John B. Conway, Subnormal Operators, Pitman, London, 1981.zbMATHGoogle Scholar
  18. [18]
    John B. Conway, The Theory of Subnormal Operators, Amer. Math. Soc. Surveys and Monographs 36, Providence, 1991.Google Scholar
  19. [19]
    John B. Conway, A Course in Operator Theory, GTM 21, Amer. Math. Soc, U.S.A., 1999.Google Scholar
  20. [20]
    Kenneth R. Davidson, Nest algebras, Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Great Britain, 1988.zbMATHGoogle Scholar
  21. [21]
    Chandler Davis, Heydar Radjavi, and Peter Rosenthal, On operator algebras and invariant subspaces, Cann. J. Math. 21 (1969), 1178–1181.CrossRefzbMATHMathSciNetGoogle Scholar
  22. [22]
    J.A. Deddens, Every isometry is reflexive, Proc. Amer. Math. Soc. 28 (1971), 509–512.CrossRefzbMATHMathSciNetGoogle Scholar
  23. [23]
    J. Drxmier, Les opérateurs permutables à l’opérateur integral, Portugal. Math. Fas. 2 8 (1949), 73–84.Google Scholar
  24. [24]
    W.F. Donoghue, The lattice of invariant subspaces of a completely continuous quasi-nilpotent transformation, Pacific J. Math.7 (1957), 1031–1035.zbMATHMathSciNetGoogle Scholar
  25. [25]
    D.W. Hadwin, E.A. Nordgren, Heydar Radjavi, and Peter Rosenthal, An operator not satisfying Lomonosov’s hypothesis, J. Func. Anal. 38 (1980), 410–415.CrossRefzbMATHMathSciNetGoogle Scholar
  26. [26]
    P.R. Halmos, Normal dilations and extensions of operators, Summa Bras. Math. II (1950), 125–134.MathSciNetGoogle Scholar
  27. [27]
    P.R. Halmos, Shifts on Hilbert spaces, J. Reine Angew. Math. 208 (1961), 102–112.CrossRefzbMATHMathSciNetGoogle Scholar
  28. [28]
    P.R. Halmos, A Glimpse into Hilbert Space, Lectures on Modern Mathematics, Vol. I, T. Saaty (Ed.), pp. 1–22, J. Wiley & Sons Inc. (1963).Google Scholar
  29. [29]
    P.R. Halmos, Invariant subspaces of polynomially compact operators, Pacific J. Math. 16, (1966), 433–437.zbMATHMathSciNetGoogle Scholar
  30. [30]
    P.R. Halmos, Irreducible operators, Michigan Math. J. 15 (1968), 215–223.CrossRefzbMATHMathSciNetGoogle Scholar
  31. [31]
    P.R. Halmos, Quasitriangular operators, Acta Scien. Math. XXIX (1968), 259–293.Google Scholar
  32. [32]
    P.R. Halmos, Ten Problems in Hilbert Space, Bull. A.M.S. 76, (1970), 887–933.CrossRefzbMATHMathSciNetGoogle Scholar
  33. [33]
    P.R. Halmos, Reflexive lattices of subspaces, J. London Math. Soc. (2) 4, (1971), 257–263.Google Scholar
  34. [34]
    P.R. Halmos, Ten years in Hilbert space, Integ. Equat. Op. Th. 2/4 (1979), 529–563.CrossRefzbMATHMathSciNetGoogle Scholar
  35. [35]
    P.R. Halmos, The Heart of Mathematics, American Mathematical Monthly 87 (1980), 519–524.CrossRefzbMATHMathSciNetGoogle Scholar
  36. [36]
    P.R. Halmos, A Hilbert Space Problem Book, Second Edition, GTM 19, Springer-Verlag, New York, 1982.Google Scholar
  37. [37]
    P.R. Halmos, Selecta: Research Contributions, Springer-Verlag, New York, 1983.zbMATHGoogle Scholar
  38. [38]
    P.R. Halmos, Selecta: Expository Writing, Springer-Verlag, New York, 1983.zbMATHGoogle Scholar
  39. [39]
    K.J. Harrison and W.E. Longstaff, An invariant subspace lattice of order type ω+ω + 1, Proc. Math. Soc. 79 (1980), 45–49.zbMATHMathSciNetGoogle Scholar
  40. [40]
    David R. Larson, Triangularity in operator algebras, Surveys of some recent results in operator theory, Vol. II, Pitman Res. Notes Math. 192, Longman, Harlow, 1988, pp. 121–188.Google Scholar
  41. [41]
    V.J. Lomonosov, Invariant subspaces for operators commuting with compact operators, Functional Anal. and Appl. 7 (1973), 55–56.MathSciNetGoogle Scholar
  42. [42]
    John E. McCarthy, Reflexivity of subnormal operators, Pacific J. Math. 161 (1993), 359–370.zbMATHMathSciNetGoogle Scholar
  43. [43]
    A.J. Michaels, Hilden’s simple proof of Lomonosov’s invariant subspace theorem, Adv. Math. 25 (1977), 56–58.CrossRefzbMATHMathSciNetGoogle Scholar
  44. [44]
    Rober F. Olin and James E. Thomson, Algebras of subnormal operators, J. Func. Anal. 37 (1980), 271–301.CrossRefzbMATHMathSciNetGoogle Scholar
  45. [45]
    Heydar Radjavi and Peter Rosenthal, On invariant subspaces and reflexive algebras, Amer. J. Math. 91 (1969), 683–692.CrossRefzbMATHMathSciNetGoogle Scholar
  46. [46]
    Heydar Radjavi and Peter Rosenthal, A sufficient condition that an operator algebra be self-adjoint, Cann. J. Math. 23 (1971), 588–597.CrossRefzbMATHMathSciNetGoogle Scholar
  47. [47]
    Heydar Radjavi and Peter Rosenthal, Invariant Subspaces, second edition, Dover, Mineola, N.Y., 2003.zbMATHGoogle Scholar
  48. [48]
    Heydar Radjavi and Peter Rosenthal, Simultaneous triangularization, Springer, New York, 2000.zbMATHGoogle Scholar
  49. [49]
    Peter Rosenthal, Examples of invariant subspace lattices, Duke Math. J. 37 (1970), 103–112.CrossRefzbMATHMathSciNetGoogle Scholar
  50. [50]
    D.E. Sarason, Invariant subspaces and unstarred operator algebras, Pacific J. Math. 17 (1966), 511–517.zbMATHMathSciNetGoogle Scholar
  51. [51]
    D.E. Sarason, Invariant subspaces, Mathematical Surveys 13 (edited by C. Pearcy), A.M.S., Providence, 1974, pp. 1–47.Google Scholar
  52. [52]
    Dan Voiculescu, A non-commutative Weyl-von Neumann theorem, Rev. Roumaine Math. Pures Appl. 21 (1976), 97–113.zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  • Heydar Radjavi
    • 1
  • Peter Rosenthal
    • 2
  1. 1.Department of MathematicsUniversity of WaterlooWaterlooCanada
  2. 2.Department of MathematicsUniversity of TorontoTorontoCanada

Personalised recommendations