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Open Access Published by De Gruyter Open Access October 23, 2012

An overview of some recent developments on the Invariant Subspace Problem

Isabelle Chalendar EMAIL logo and Jonathan R. Partington
From the journal Concrete Operators

An overview of some recent developments on the Invariant Subspace Problem

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.

References

Ambrozie C., Müller V., Invariant subspaces for polynomially bounded operators. J. Funct. Anal., 2004, 213, 321–34510.1016/j.jfa.2003.12.004Search in Google Scholar

Androulakis G., Dodos P., Sirotkin G., Troitsky V.G., Classes of strictly singular operators and their products, Israel J. Math., (to appear)Search in Google Scholar

Anisca R., Troitsky V.G., Minimal vectors of positive operators, Indiana Univ. Math. J., 2005, 54, 861–87210.1512/iumj.2005.54.2544Search in Google Scholar

Ansari S., Enflo P., Extremal vectors and invariant subspaces, Trans. Amer. Math. Soc., 1998, 350, 539–55810.1090/S0002-9947-98-01865-0Search in Google Scholar

Argyros S.A., Haydon R.G., A hereditarily indecomposable L-space that solves the scalar-plus-compact problem, Acta Math., 2011, 206, 1–54 10.1007/s11511-011-0058-ySearch in Google Scholar

Aronszajn N., Smith K.T., Invariant subspaces of completely continuous operators, Ann. of Math. (2), 1954, 60, 345–35010.2307/1969637Search in Google Scholar

Atzmon A., An operator without invariant subspace on a nuclear Fréchet space, Ann. of Math. (2), 1983, 117, 669–69410.2307/2007039Search in Google Scholar

Beauzamy B., Un opérateur sans sous-espace invariant: simplification de l’exemple de P. Enflo, Integral Equations Operator Theory, 1985, 8, 314–38410.1007/BF01202903Search in Google Scholar

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 10.1090/cbms/056Search in Google Scholar

Bercovici H., Foias C., Pearcy C., Two Banach space methods and dual operator algebras, J. Funct. Anal., 1988, 78, 306–345 10.1016/0022-1236(88)90122-XSearch in Google Scholar

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 10.2140/pjm.1966.16.421Search in Google Scholar

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 Search in Google Scholar

Brown S.W., Some invariant subspaces for subnormal operators, Integral Equations Operator Theory, 1978, 1, 310–333 10.1007/BF01682842Search in Google Scholar

Brown S.W., Chevreau B., Pearcy C., On the structure of contraction operators II, J. Funct. Anal., 1988, 76, 30–55 10.1016/0022-1236(88)90047-XSearch in Google Scholar

Caradus S.R., Universal operators and invariant subspaces, Proc. Amer. Math. Soc., 1969, 23, 526–527 10.1090/S0002-9939-1969-0250104-4Search in Google Scholar

Casazza P., Lohman R., A general construction of spaces of the type of R. C. James, Canad. J. Math., 1975, 27, 1263–1270 10.4153/CJM-1975-131-3Search in Google Scholar

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 10.1016/j.jfa.2008.09.010Search in Google Scholar

Chalendar I., Partington J.R., Convergence properties of minimal vectors for normal operators and weighted shifts, Proc. Amer. Math. Soc., 2005, 133, 501–510 10.1090/S0002-9939-04-07595-1Search in Google Scholar

Chalendar I., Partington J.R., Variations on Lomonosov’s theorem via the technique of minimal vectors, Acta Sci. Math. (Szeged), 2005, 71, 603–617 Search in Google Scholar

Chalendar I., Partington J.R., Invariant subspaces for products of Bishop operators, Acta Sci. Math. (Szeged), 2008, 74, 719–727 Search in Google Scholar

Chalendar I., Partington J.R., Modern approaches to the invariant-subspace problem, Cambridge Tracts in Mathematics, 188, Cambridge University Press, Cambridge, 2011 10.1017/CBO9780511862434Search in Google Scholar

Cima J.A., Thomson J., Wogen W., On some properties of composition operators, Indiana Univ. Math. J., 1974/75, 24, 215–220 10.1512/iumj.1975.24.24018Search in Google Scholar

Davie A.M., Invariant subspaces for Bishop’s operators, Bull. London Math. Soc., 1974, 6, 343–348 10.1112/blms/6.3.343Search in Google Scholar

Delpech S., A short proof of Pitt’s compactness theorem, Proc. Amer. Math. Soc., 2009, 137, 1371–1372 10.1090/S0002-9939-08-09617-2Search in Google Scholar

Enflo P., On the invariant subspace problem in Banach spaces, Séminaire Maurey–Schwartz (1975–1976) Espaces Lp, applications radonifiantes et géométrie des espaces de Banach, Exp. Nos. 14–15, 7 pp., Centre Math., École Polytech., Palaiseau, 1976 Search in Google Scholar

Enflo P., On the invariant subspace problem for Banach spaces, Acta Math., 1987, 158, 213–313 10.1007/BF02392260Search in Google Scholar

Enflo P., Extremal vectors for a class of linear operators, Functional analysis and economic theory (Samos, 1996), 61–64, Springer, Berlin, 1998 10.1007/978-3-642-72222-6_5Search in Google Scholar

Enflo P., Hõim T., Some results on extremal vectors and invariant subspaces, Proc. Amer. Math. Soc., 2003, 131, 379–387 10.1090/S0002-9939-02-06326-8Search in Google Scholar

Fabian M., Halala P., Hájek P., Montesinos Santalucía V., Pelant J., Zizler V., Functional analysis and infinite geometry, CMS Books in Mathematics, Springer-Verlag, New-York, 2001 10.1007/978-1-4757-3480-5Search in Google Scholar

Flattot A., Hyperinvariant subspaces for Bishop-type operators, Acta Sci. Math. (Szeged), 2008, 74, 689–718 Search in Google Scholar

Fulton W., Algebraic topology, Springer–Verlag, New York, 1995 10.1007/978-1-4612-4180-5Search in Google Scholar

Gallardo-Gutiérrez E.A., Gorkin P., Minimal invariant subspaces for composition operators, J. Math. Pures Appl. (9), 2011, 95, 245–259 10.1016/j.matpur.2010.04.003Search in Google Scholar

Gowers W.T., Maurey B., Banach spaces with small spaces of operators, Math. Ann., 1997, 307, 543–568 10.1007/s002080050050Search in Google Scholar

Grünbaum B., Convex polytopes, 2nd edition, Graduate Texts in Mathematics, 221, Springer–Verlag, New York, 2003 10.1007/978-1-4613-0019-9Search in Google Scholar

James R.C., A non-reflexive Banach space isometric with its second conjugate, Proc. Nat. Acad. Sci. U.S.A., 1951, 37, 174–177 10.1073/pnas.37.3.174Search in Google Scholar PubMed PubMed Central

Kim H.J., Hyperinvariant subspaces for operators having a normal part, Oper. Matrices, 2011, 5, 487–494 10.7153/oam-05-36Search in Google Scholar

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 10.1007/3-7643-7314-8_9Search in Google Scholar

Lindenstrauss J., Tzafriri L., Classical Banach spaces. I, Springer-Verlag, Berlin, 1977 10.1007/978-3-642-66557-8Search in Google Scholar

Littlewood J.E., On inequalities in the theory of functions, Proc. London Math. Soc. (2), 1925, 23, 481–519 10.1112/plms/s2-23.1.481Search in Google Scholar

Lomonosov V.I., Invariant subspaces for operators commuting with compact operators, Funct. Anal. Appl., 1973, 7, 213–214 10.1007/BF01080698Search in Google Scholar

MacDonald G.W., Invariant subspaces for Bishop-type operators, J. Funct. Anal., 1990, 91, 287–311 10.1016/0022-1236(90)90146-CSearch in Google Scholar

Maslyuchenko V., Plichko A., Quasireflexive locally convex spaces without Banach subspaces, Teor. Funktsiˇı Funktsional. Anal. i Prilozhen., 1985, 44, 78–84 (in Russian), translation in J. Soviet Math., 1990, 48, 307–312 10.1007/BF01101251Search in Google Scholar

Matache V., On the minimal invariant subspaces of the hyperbolic composition operator, Proc. Amer. Math. Soc., 1993, 119, 837–841 10.1090/S0002-9939-1993-1152988-8Search in Google Scholar

Milman V.D., Operators of class C0 and C0*, Teor. Funktsiˇı Funkcional. Anal. i Priložen., 1970, 10, 15–26 Search in Google Scholar

Mortini R., Cyclic subspaces and eigenvectors of the hyperbolic composition operator, Travaux mathématiques, Fasc. VII, 69–79, Sém. Math. Luxembourg, Centre Univ. Luxembourg, Luxembourg, 1995 Search in Google Scholar

Nordgren E., Rosenthal P., Wintrobe F.S., Invertible composition operators on Hp, J. Funct. Anal., 1987, 73, 324–344 10.1016/0022-1236(87)90071-1Search in Google Scholar

Partington J.R., Pozzi E., Universal shifts and composition operators, Oper. Matrices, 2011, 5, 455–467 10.7153/oam-05-33Search in Google Scholar

Pisier G., A polynomially bounded operator on Hilbert space which is not similar to a contraction, J. Amer. Math. Soc., 1997, 10, 351–369 10.1090/S0894-0347-97-00227-0Search in Google Scholar

Plichko A., Superstrictly singular and superstrictly cosingular operators, Functional analysis and its applications, 2004, North-Holland Math. Stud., 197, Elsevier, Amsterdam, 239–255 10.1016/S0304-0208(04)80172-5Search in Google Scholar

Popov A.I., Schreier singular operators, Houston J. Math., 2009, 35, 209–222 Search in Google Scholar

Pozzi E., Universality of weighted composition operators on L2([0, 1]) and Sobolev spaces, Acta Sci. Math. (Szeged), (to appear) Search in Google Scholar

Read C., A solution to the invariant subspace problem, Bull. London Math. Soc., 1984, 16, 337–401 10.1112/blms/16.4.337Search in Google Scholar

Read C., A solution to the invariant subspace problem on the space l1, Bull. London Math. Soc., 1985, 17, 305–317 10.1112/blms/17.4.305Search in Google Scholar

Read C., A short proof concerning the invariant subspace problem, J. London Math. Soc. (2), 1986, 34, 335–348 10.1112/jlms/s2-34.2.335Search in Google Scholar

Read C., Quasinilpotent operators and the invariant subspace problem, J. London Math. Soc. (2), 1997, 56, 595–606 10.1112/S0024610797005486Search in Google Scholar

Read C., Strictly singular operators and the invariant subspace problem, Studia Math., 1999, 132, 203–226 10.4064/sm-132-3-203-226Search in Google Scholar

Singer I., Bases in Banach Spaces I, Springer-Verlag, New York–Berlin, 1970 10.1007/978-3-642-51633-7Search in Google Scholar

Sari B., Schlumprecht Th., Tomczak-Jaegermann N., Troitsky V.G., On norm closed ideals in L(lp⊗ lq), Studia Math., 2007, 179, 239–262 10.4064/sm179-3-3Search in Google Scholar

Thomson J.E., Invariant subspaces for algebras of subnormal operators, Proc. Amer. Math. Soc., 1986, 96, 462–464 10.1090/S0002-9939-1986-0822440-1Search in Google Scholar

Troitsky V.G., Lomonosov’s theorem cannot be extended to chains of four operators, Proc. Amer. Math. Soc., 2000, 128, 521–525 10.1090/S0002-9939-99-05176-XSearch in Google Scholar

Troitsky V.G., Minimal vectors in arbitrary Banach spaces, Proc. Amer. Math. Soc., 2004, 132, 1177–1180 10.1090/S0002-9939-03-07223-XSearch in Google Scholar

Ziegler G.M., Lectures on polytopes, Graduate Texts in Mathematics, 152, Springer-Verlag, New York, 199410.1007/978-1-4613-8431-1Search in Google Scholar

Published Online: 2012-10-23

©2012 Versita Sp. z o.o.

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