Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Minimal vectors in arbitrary Banach spaces


Author: Vladimir G. Troitsky
Journal: Proc. Amer. Math. Soc. 132 (2004), 1177-1180
MSC (2000): Primary 47A15
DOI: https://doi.org/10.1090/S0002-9939-03-07223-X
Published electronically: August 28, 2003
MathSciNet review: 2045435
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We extend the method of minimal vectors to arbitrary Banach spaces. It is proved, by a variant of the method, that certain quasinilpotent operators on arbitrary Banach spaces have hyperinvariant subspaces.


References [Enhancements On Off] (What's this?)

  • [AB99] C. D. Aliprantis and K. C. Border, Infinite-dimensional analysis, a hitchhiker's guide, second edition, Springer-Verlag, Berlin, 1999. MR 2000k:46001
  • [AE98] S. Ansari and P. Enflo, Extremal vectors and invariant subspaces, Trans. Amer. Math. Soc. 350 (1998), no. 2, 539-558. MR 98d:47019
  • [A] G. Androulakis, A note on the method of minimal vectors, Trends in Banach spaces and operator theory, Contemporary Mathematics (A. Kaminska, editor), Vol. 321, Amer. Math. Soc., Providence, RI, 2003, pp. 29-36.
  • [CPS] I. Chalendar, J. R. Partington, and M. Smith, Approximation in reflexive Banach spaces and applications to the invariant subspace problem, Proc. Amer. Math. Soc. (2004), to appear.
  • [E76] P. Enflo, On the invariant subspace problem in Banach spaces, Séminaire Maurey-Schwartz (1975-1976) Espaces $L\sp{p}$, applications radonifiantes et géométrie des espaces de Banach, Exp. Nos. 14-15, Centre Math., École Polytech., Palaiseau, 1976, pp. 1-7. MR 57:13530
  • [E87] -, On the invariant subspace problem for Banach spaces, Acta Math. 158 (1987), no. 3-4, 213-313. MR 88j:47006
  • [H81] N. D. Hooker, Lomonosov's hyperinvariant subspace theorem for real spaces, Math. Proc. Cambridge Philos. Soc. 89 (1981), no. 1, 129-133. MR 84a:47009
  • [L73] V. I. Lomonosov, Invariant subspaces of the family of operators that commute with a completely continuous operator, Funkcional. Anal. i Prilozen. 7 (1973), no. 3, 55-56. MR 54:8319
  • [P] C. Pearcy, On technique of Enflo, Proc. Amer. Math. Soc., to appear.
  • [R84] C. J. Read, A solution to the invariant subspace problem, Bull. London Math. Soc. 16 (1984), no. 4, 337-401. MR 86f:47005
  • [R85] -, A solution to the invariant subspace problem on the space $\ell\sb 1$, Bull. London Math. Soc. 17 (1985), no. 4, 305-317. MR 87e:47013
  • [RR73] H. Radjavi and P. Rosenthal, Invariant subspaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 77, Springer-Verlag, New York, 1973. MR 51:3924

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47A15

Retrieve articles in all journals with MSC (2000): 47A15


Additional Information

Vladimir G. Troitsky
Affiliation: Department of Mathematics, University of Alberta, Edmonton, AB, T6G 2G1 Canada
Email: vtroitsky@math.ualberta.ca

DOI: https://doi.org/10.1090/S0002-9939-03-07223-X
Keywords: Invariant subspace, minimal vector
Received by editor(s): November 27, 2002
Received by editor(s) in revised form: December 22, 2002
Published electronically: August 28, 2003
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society