On quasi-affine transforms of Read’s operator
HTML articles powered by AMS MathViewer
- by Thomas Schlumprecht and Vladimir G. Troitsky
- Proc. Amer. Math. Soc. 131 (2003), 1405-1413
- DOI: https://doi.org/10.1090/S0002-9939-02-06896-X
- Published electronically: December 6, 2002
- PDF | Request permission
Abstract:
We show that C. J. Read’s example of an operator $T$ on $\ell _1$ which does not have any non-trivial invariant subspaces is not the adjoint of an operator on a predual of $\ell _1$. Furthermore, we present a bounded diagonal operator $D$ such that even though $D^{-1}$ is unbounded, the operator $D^{-1}TD$ is a bounded operator on $\ell _1$ with invariant subspaces, and is adjoint to an operator on $c_0$.References
- Y. A. Abramovich, C. D. Aliprantis, and O. Burkinshaw, The invariant subspace problem: some recent advances, Rend. Istit. Mat. Univ. Trieste 29 (1998), no. suppl., 3–79 (1999). Workshop on Measure Theory and Real Analysis (Italian) (Grado, 1995). MR 1696022
- P. Enflo, On the invariant subspace problem in Banach spaces, Séminaire Maurey-Schwartz (1975–1976): Espaces $L^p$, applications radonifiantes et géométrie des espaces de Banach, Centre Math., École Polytech., Palaiseau, 1976, pp. Exp. Nos. 14-15, 7. MR 0473871
- Per Enflo, On the invariant subspace problem for Banach spaces, Acta Math. 158 (1987), no. 3-4, 213–313. MR 892591, DOI 10.1007/BF02392260
- C. J. Read, A solution to the invariant subspace problem on the space $l_1$, Bull. London Math. Soc. 17 (1985), no. 4, 305–317. MR 806634, DOI 10.1112/blms/17.4.305
- C. J. Read, A short proof concerning the invariant subspace problem, J. London Math. Soc. (2) 34 (1986), no. 2, 335–348. MR 856516, DOI 10.1112/jlms/s2-34.2.335
- Heydar Radjavi and Peter Rosenthal, Invariant subspaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 77, Springer-Verlag, New York-Heidelberg, 1973. MR 0367682, DOI 10.1007/978-3-642-65574-6
- Béla Sz.-Nagy and Ciprian Foiaş, Vecteurs cycliques et quasi-affinités, Studia Math. 31 (1968), 35–42 (French). MR 236756, DOI 10.4064/sm-31-1-35-42
Bibliographic Information
- Thomas Schlumprecht
- Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
- MR Author ID: 260001
- Email: schlump@math.tamu.edu
- Vladimir G. Troitsky
- Affiliation: Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
- Email: vtroitsky@math.ualberta.ca
- Received by editor(s): November 30, 2001
- Published electronically: December 6, 2002
- Additional Notes: The first author was supported by the NSF. Most of the work on the paper was done during the Workshop on linear analysis and probability at Texas A&M University, College Station
- Communicated by: David R. Larson
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 1405-1413
- MSC (2000): Primary 47A15; Secondary 47B37
- DOI: https://doi.org/10.1090/S0002-9939-02-06896-X
- MathSciNet review: 1949870