Strictly singular operators and the invariant subspace problem
Properties of strictly singular operators have recently become of topical interest because the work of Gowers and Maurey in [GM1] and [GM2] gives (among many other brilliant and surprising results, such as those in [G1] and [G2]) Banach spaces on which every continuous operator is of form λ I + S, where S is strictly singular. So if strictly singular operators had invariant subspaces, such spaces would have the property that all operators on them had invariant subspaces. However, in this paper we exhibit examples of strictly singular operators without nontrivial closed invariant subspaces. So, though it may be true that operators on the spaces of Gowers and Maurey have invariant subspaces, yet this cannot be because of a general result about strictly singular operators. The general assertion about strictly singular operators is false.
- [E1] P. Enflo, On the invariant subspace problem for Banach spaces, Acta Math. 158 (1987), 212-313.
- [G1] W. T. Gowers, A solution to Banach's hyperplane problem, Bull. London Math. Soc. 26 (1994), 523-530.
- [G2] W. T. Gowers, A solution to the Schroeder-Bernstein problem for Banach spaces, ibid. 28 (1996), 297-304.
- [GM1] W. T. Gowers and B. Maurey, The unconditional basic sequence problem, J. Amer. Math. Soc. 16 (1993), 851-874.
- [GM2] W. T. Gowers and B. Maurey, Banach spaces with small spaces of operators, Math. Ann. 307 (1997), 543-568.
- [J1] R. C. James, A non-reflexive Banach space isometric with its second conjugate space, Proc. Nat. Acad. Sci. U.S.A. 37 (1951), 174-177.
- [R1] C. J. Read, A solution to the invariant subspace problem, Bull. London Math. Soc. 16 (1984), 337-401.
- [R2] C. J. Read, A short proof concerning the invariant subspace problem, J. London Math. Soc. (2) 33 (1986), 335-348.
- [R3] C. J. Read, The invariant subspace problem, a description with further applications of a combinatorial proof, in: Advances in Invariant Subspaces and Other Results of Operator Theory, Birkhäuser, 1986, 275-300.
- [R4] C. J. Read, A solution to the invariant subspace problem on the space $l_1$, Bull. London Math. Soc. 17 (1985), 305-317.
- [R5] C. J. Read, The invariant subspace problem on a class of nonreflexive Banach spaces, in: Geometric Aspects of Functional Analysis, Israel Seminar (GAFA) 1986-7, Lecture Notes in Math. 1317, Springer, 1988, 1-20.
- [R6] C. J. Read, The invariant subspace problem for a class of Banach spaces, 2: Hypercyclic operators, Israel J. Math. 63 (1988), 1-40.
- [R7] C. J. Read, The invariant subspace problem on some Banach spaces with separable dual, Proc. London Math. Soc. (3) 58 (1989), 583-607.
- [R8] C. J. Read, Quasinilpotent operators and the invariant subspace problem, J. London Math. Soc. (2) 56 (1997), 595-606.
- [Ra1] H. Radjavi and P. Rosenthal, Invariant Subspaces, Springer, 1973.