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An overview of some recent developments on the Invariant Subspace Problem

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Abstrakty
EN
This paper presents an account of some recent approaches to the Invariant Subspace Problem. It contains a brief historical account of the problem, and some more detailed discussions of specific topics, namely, universal operators, the Bishop operators, and Read’s Banach space counter-example involving a finitely strictly singular operator.
Twórcy
  • Université de Lyon; CNRS; Université Lyon 1; INSA de Lyon; Ecole Centrale de Lyon, CNRS, UMR 5208, Institut Camille Jordan 43 bld. du 11 novembre 1918, F-69622 Villeurbanne Cedex, France
  • School of Mathematics, University of Leeds, Leeds LS2 9JT, U.K.
Bibliografia
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  • Anisca R., Troitsky V.G., Minimal vectors of positive operators, Indiana Univ. Math. J., 2005, 54, 861–872[Crossref]
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  • Beauzamy B., Un opérateur sans sous-espace invariant: simplification de l’exemple de P. Enflo, Integral Equations Operator Theory, 1985, 8, 314–384
  • Bercovici H., Foias C., Pearcy C., Dual algebras with applications to invariant subspaces and dilation theory, CBMS Regional conference series in mathematics, 56. A.M.S., Providence, 1985
  • Bercovici H., Foias C., Pearcy C., Two Banach space methods and dual operator algebras, J. Funct. Anal., 1988, 78, 306–345 [Crossref]
  • Bernstein A.R., Robinson A., Solution of an invariant subspace problem of K. T. Smith and P. R. Halmos, Pacific J. Math., 1966, 16, 421–431
  • Blecher D.P., Davie A.M., Invariant subspaces for an operator on L2 (Π) composed of a multiplication and a translation, J. Operator Theory, 1990, 23, 115–123
  • Brown S.W., Some invariant subspaces for subnormal operators, Integral Equations Operator Theory, 1978, 1, 310–333
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  • Chalendar I., Fricain E., Popov A.I., Timotin D., Troitsky V.G., Finitely strictly singular operators between James spaces, J. Funct. Anal., 2009, 256, 1258–1268
  • Chalendar I., Partington J.R., Convergence properties of minimal vectors for normal operators and weighted shifts, Proc. Amer. Math. Soc., 2005, 133, 501–510
  • Chalendar I., Partington J.R., Variations on Lomonosov’s theorem via the technique of minimal vectors, Acta Sci. Math. (Szeged), 2005, 71, 603–617
  • Chalendar I., Partington J.R., Invariant subspaces for products of Bishop operators, Acta Sci. Math. (Szeged), 2008, 74, 719–727
  • Chalendar I., Partington J.R., Modern approaches to the invariant-subspace problem, Cambridge Tracts in Mathematics, 188, Cambridge University Press, Cambridge, 2011
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  • Enflo P., On the invariant subspace problem for Banach spaces, Acta Math., 1987, 158, 213–313
  • Enflo P., Extremal vectors for a class of linear operators, Functional analysis and economic theory (Samos, 1996), 61–64, Springer, Berlin, 1998
  • Enflo P., Hõim T., Some results on extremal vectors and invariant subspaces, Proc. Amer. Math. Soc., 2003, 131, 379–387
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  • Flattot A., Hyperinvariant subspaces for Bishop-type operators, Acta Sci. Math. (Szeged), 2008, 74, 689–718
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  • Kim H.J., Hyperinvariant subspaces for operators having a normal part, Oper. Matrices, 2011, 5, 487–494
  • Kumar R., Partington J.R., Weighted composition operators on Hardy and Bergman spaces, Recent advances in operator theory, operator algebras, and their applications, 157–167, Oper. Theory Adv. Appl., 153, Birkhäuser, Basel, 2005
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  • MacDonald G.W., Invariant subspaces for Bishop-type operators, J. Funct. Anal., 1990, 91, 287–311
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  • Partington J.R., Pozzi E., Universal shifts and composition operators, Oper. Matrices, 2011, 5, 455–467
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  • Plichko A., Superstrictly singular and superstrictly cosingular operators, Functional analysis and its applications, 2004, North-Holland Math. Stud., 197, Elsevier, Amsterdam, 239–255
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  • Pozzi E., Universality of weighted composition operators on L2([0, 1]) and Sobolev spaces, Acta Sci. Math. (Szeged), (to appear)
  • Read C., A solution to the invariant subspace problem, Bull. London Math. Soc., 1984, 16, 337–401 [Crossref]
  • Read C., A solution to the invariant subspace problem on the space l1, Bull. London Math. Soc., 1985, 17, 305–317 [Crossref]
  • Read C., A short proof concerning the invariant subspace problem, J. London Math. Soc. (2), 1986, 34, 335–348
  • Read C., Quasinilpotent operators and the invariant subspace problem, J. London Math. Soc. (2), 1997, 56, 595–606 [Crossref]
  • Read C., Strictly singular operators and the invariant subspace problem, Studia Math., 1999, 132, 203–226
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  • Thomson J.E., Invariant subspaces for algebras of subnormal operators, Proc. Amer. Math. Soc., 1986, 96, 462–464 [Crossref]
  • Troitsky V.G., Lomonosov’s theorem cannot be extended to chains of four operators, Proc. Amer. Math. Soc., 2000, 128, 521–525
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  • Ziegler G.M., Lectures on polytopes, Graduate Texts in Mathematics, 152, Springer-Verlag, New York, 1994
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_conop-2012-0001
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