Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-14T07:14:07.071Z Has data issue: false hasContentIssue false
  • English
  • Français

Some Recent Results on Invariant Subspaces

Published online by Cambridge University Press:  20 November 2018

Peter Rosenthal
Peter Rosenthal*
Affiliation:
Department of Mathematics, University of TorontoToronto, OntarioM5S 1A1
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This expository paper surveys work on invariant subspaces and related topics which has been done in the past few years. We recommend, naturally, that the reader consult [52] for work done prior to 1973 and [54] for a discussion of some of the consequences of Lomonosov’s Lemma; (Lomonosov’s paper has now also appeared in English ([36])).

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 19 , Issue 3 , 01 September 1976 , pp. 303 - 313
Copyright
Copyright © Canadian Mathematical Society 1976

References

1. Abrahamse, M. B. and Douglas, R. G., A class of subnormal operators related to multiply-connected domains, Advances in Math. 19 (1976), 106–148.Google Scholar
2. Apostol, C., On the growth of resolvent, perturbation, and invariant subspaces, Rev. Roumaine Math. Pures Appl. 16 (1971), 161–172.Google Scholar
3. Apostol, C., Douglas, R. G. and Foias, C., Quasi-similar models for nilpotent operators, to appear.Google Scholar
4. Apostol, C., Foias, C., and Voiculescu, D., Some results on non-quasitriangular operators IV, Rev. Roumaine Math. Pures Appl. 18 (1973), 487–514.Google Scholar
5. Apostol, C., Foias, C., and Voiculescu, D., Strongly reductive operators are normal, Acta Sci. Math. (Szeged), to appear.Google Scholar
6. Apostol, C., Foias, C., and Voiculescu, D., On strongly reductive algebras, Rev. Roumaine Math. Pures Appl., to appear.Google Scholar
7. Apostol, C. and Fong, C. K., Invariant subspaces for algebras generated by strongly reductiveoperators, Duke Math. J. 42 (1975), 495–498.Google Scholar
8. Arveson, W. B., Operator algebras and invariant subspaces, Annals of Math. 100 (1974), 433–532.Google Scholar
9. Azoff, E., Fong, K. C., and Gilfeather, F., A reduction theory for non-self-adjoint operatoralgebras, Trans. Amer. Math. Soc, to appear.Google Scholar
10. Bonsall, F. F. and Rosenthal, Peter, Certain Jordan operator algebras and double commutanttheorems, J. Functional Anal. 21 (1976), 155–186.Google Scholar
11. Brown, L. G., Douglas, R. G., and Fillmore, P. A., Unitary equivalence modulo the compactoperators and extensions of -algebras, in Proceedings of a Conference on Operator Theory, Springer-Verlag Lecture Notes in Mathematics, vol. 345, 1973, pp. 58–128.Google Scholar
12. Brown, L. G., Douglas, R. G., and Fillmore, P. A., Extensions of -algebras and Khomology, to appear.Google Scholar
13. Clancey, K. and Moore, B. III, Operators of class and transitive algebras, Acta Sci. Math. (Szeged) 36 (1974), 215–218.Google Scholar
14. Colojoara, I. and Foias, C., Theory of Generalized Spectral Operators, Gordan and Breach, New York, 1968.Google Scholar
15. Conway, John B. and Yuan, Wu Pei, The splitting of and related questions, to appear.Google Scholar
16. Conway, John B. and Olin, Robert F., A functional calculus for subnormal operators II, to appear.Google Scholar
17. Daughtry, John, An invariant subspace theorem, Proc. Amer. Math. Soc. 49 (1975), 267–268.Google Scholar
18. Davie, A. M., Invariant subspaces for Bishop’s operators, Bull. Lond. Math. Soc. 6 (1974), 343–348.Google Scholar
19. Davis, Chandler, Generators of the ring of bounded operators, Proc. Amer. Math. Soc. 6 (1955), 970–972.Google Scholar
20. Deddens, J. and Wogen, W., On operators with the double commutant property, to appeal.Google Scholar
21. Douglas, R. G. and Pearcy, Carl, Invariant subspaces of non-quasitriangular operators, in Proceedings of a Conference on Operator Theory, Springer-Verlag lecture Notes in Mathematics, vol. 345, 1973, pp. 13–57.Google Scholar
22. Dyer, J. and Porcelli, P., Concerning the invariant subspace problem, Notices Amer. Math. Soc. 17 (1970), 788.Google Scholar
23. Dyer, J., Pedersen, E., and Porcelli, P., An equivalent formulation of the invariant subspaceproblem, Bull. Amer. Math. Soc. 78 (1972), 1020–1023.Google Scholar
24. Foias, C., Invariant para-closed subspaces, Indiana Univ. Math. J. 21 (1972), 887–906.Google Scholar
25. Fong, C. K., A sufficient condition that an operator be normal, Michigan Math. J. 21 (1974), 161–162.Google Scholar
26. Fong, C. K., On operators with reducing hyperinvariant subspaces, to appear.Google Scholar
27. Fong, C. K., Applications of direct integrals to operator theory, dissertation, University of Toronto, 1976.Google Scholar
28. Gilfeather, F., Operator valued roots of abelian analytic functions, Pac. J. Math., to appear.Google Scholar
29. Harrison, K. J., Strongly reductive operators, Acta Sci. Math. (Szeged) 37 (1975), 205–212.Google Scholar
30. Herrero, D. A., Transitive algebras containing almost unitary -contractions, to appear.Google Scholar
31. Jafarian, A. A., Existence of hyperinvariant subspaces, Indiana Univ. Math. J. 24 (1974), 565–575.Google Scholar
32. Jafarian, A. A., On reductive operators, Indiana Univ. Math. J. 23 (1974), 607–613.Google Scholar
33. Jafarian, A. A., Weak contractions of Sz.-Nagy and Foias are decomposable, Rev. Roumaine Math. Pures Appl., to appear.Google Scholar
34. Kitano, K., The growth of the resolvent and hyperinvariant subspaces, Tohoku Math. J. 25 (1973), 317–331.Google Scholar
35. Loebl, R. and Muhly, P., Analyticity and flows in von Neumann algebras, to appear.Google Scholar
36. Lomonosov, V. I., Invariant subspaces for the family of operators which commute with acompletely continuous operator, Funct. Anal, and Appl. 7 (1973), 213–214.Google Scholar
37. Moore, R. L., Reductive operators on Hilbert space, dissertation, Indiana University, 1974.Google Scholar
38. Moore, B. III and Nordgren, E., On transitive algebras containing operators, Indiana Univ. Math. J. 24 (1975), 777–784.Google Scholar
39. Muhly, P., Compact operators in the commutant of a contraction, J. Funct. Anal. 8 (1971), 197–224.Google Scholar
40. Nordgren, E., Compact operators in the algebra generated by essentially unitary operators, Proc. Amer. Math. Soc. 51 (1975), 159–162.Google Scholar
41. Nordgren, E., Rajabalipour, M., Radjavi, H., and Rosenthal, P., Algebras intertwining compactoperators, Acta Sci. Math. (Szeged), to appear.Google Scholar
42. Nordgren, E., Radjavi, H., and Rosenthal, P., On operators with reducing invariant subspaces, Amer. J. Math. XCVI (1975), 559–570.Google Scholar
43. Nordgren, E., Radjavi, H., and Rosenthal, P., A geometric equivalent of the invariant subspaceproblem, Proc. Amer. Math. Soc, to appear.Google Scholar
44. Nordgren, E., Radjavi, H., and Rosenthal, P., On Arveson’s characterization of hyperreducibletriangular algebras, Indiana Univ. Math. J., to appear.Google Scholar
45. Nordgren, E., Radjavi, H., and Rosenthal, P., Operator algebras leaving compact operatorranges invariant, Mich. Math. J., to appear.Google Scholar
46. Carl Pearcy and Shields, A. L., A survey of the Lomonosov technique in the theory of invariantsubspaces, in Topics in Operator Theory, Math. Surveys 13, Amer. Math. Soc. Providence, 1974, pp. 221–228.Google Scholar
47. Radjabalipour, M, Growth conditions, spectral operators, and reductive operators, Indiana Univ. Math. J. 23 (1974), 981–990.Google Scholar
48. Radjabalipour, M., On decomposability of compact perturbations of operators, Proc. Arner. Math. Soc. 53 (1975), 159–164.Google Scholar
49. Radjabalipour, M. and Radjavi, H., On decomposability of compact perturbations of normaloperators, Can. J. Math. XXVII (1975), 725–735.Google Scholar
50. Radjabalipour, M. and Radjavi, H., On Invariant subspaces of compact perturbations ofoperators, Rev. Roumaine Math. Pures Appl., to appear.Google Scholar
51. Radjavi, H., Reductive algebras with minimal ideals, Math. Annalen 219 (1976), 227–231.Google Scholar
52. Radjavi, H. and Rosenthal, P., Invariant Subspaces, Springer-Verlag, Berlin, 1973.Google Scholar
53. Rosenthal, P., On commutants of reductive operator algebras, Duke Math. J. 41 (1974), 829–834.Google Scholar
54. Rosenthal, P., Applications of Lomonosov’s lemma to non-self-adjoint operator algebras, Proc. Royal Irish Acad. 74 (1974), 271–281.Google Scholar
55. Rosenthal, P. and Sourour, A. R., On operator algebras containing cyclic Boolean algebras, to appear.Google Scholar
56. Sarason, D., Invariant subspaces and unstarred operator algebras, Pac. J. Math. 17 (1966), 511–517.Google Scholar
57. Sz.-Nagy, B. and Foias, C., Harmonic Analysis of Operators on Hilbert Space, Budapest, Akademiai Kiado, 1970.Google Scholar
58. Voiculescu, D., A non-commutative Weyl-von Neumann theorem, Rev. Roumaine Math. Pure Appl. 21 (1976), to appear.Google Scholar