Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
polynomially bounded operator; invariant subspace
polynomially bounded operator; invariant subspace
Summary:
We discuss the invariant subspace problem of polynomially bounded operators on a Banach space and obtain an invariant subspace theorem for polynomially bounded operators. At the same time, we state two open problems, which are relative propositions of this invariant subspace theorem. By means of the two relative propositions (if they are true), together with the result of this paper and the result of C. Ambrozie and V. Müller (2004) one can obtain an important conclusion that every polynomially bounded operator on a Banach space whose spectrum contains the unit circle has a nontrivial invariant closed subspace. This conclusion can generalize remarkably the famous result that every contraction on a Hilbert space whose spectrum contains the unit circle has a nontrivial invariant closed subspace (1988 and 1997).
References:
[1] Ambrozie, C., Müller, V.: Invariant subspaces for polynomially bounded operators. J. Funct. Anal. 213 (2004), 321-345. DOI 10.1016/j.jfa.2003.12.004 | MR 2078629 | Zbl 1056.47006
[2] Beauzamy, B.: Introduction to Operator Theory and Invariant Subspaces. North-Holland Mathematical Library 42, North-Holland, Amsterdam (1988). MR 0967989 | Zbl 0663.47002
[3] Brown, S. W., Chevreau, B., Pearcy, C.: On the structure of contraction operators. II. J. Funct. Anal. 76 (1988), 30-55. DOI 10.1016/0022-1236(88)90047-X | MR 0923043 | Zbl 0641.47013
[4] Chalendar, I., Partington, J. R.: Modern Approaches to the Invariant-Subspace Problem. Cambridge Tracts in Mathematics 188, Cambridge University Press, Cambridge (2011). MR 2841051 | Zbl 1231.47005
[5] Jiang, J.: Bounded Operators without Invariant Subspaces on Certain Banach Spaces. Thesis (Ph.D.), The University of Texas at Austin, ProQuest LLC, Ann Arbor (2001). MR 2702823
[6] Laursen, K. B., Neumann, M. M.: An Introduction to Local Spectral Theory. London Mathematical Society Monographs. New Series 20, Clarendon Press, Oxford (2000). MR 1747914 | Zbl 0957.47004
[7] Lomonosov, V.: An extension of Burnside's theorem to infinite-dimensional spaces. Isr. J. Math. 75 (1991), 329-339. DOI 10.1007/BF02776031 | MR 1164597 | Zbl 0777.47005
[8] Pisier, G.: A polynomially bounded operator on Hilbert space which is not similar to a contraction. J. Am. Math. Soc. 10 (1997), 351-369. DOI 10.1090/S0894-0347-97-00227-0 | MR 1415321 | Zbl 0869.47014
[9] Rudin, W.: Function Theory in the Unit Ball of $C^n$. Grundlehren der mathematischen Wissenschaften 241, Springer, Berlin (1980). DOI 10.1007/978-3-540-68276-9 | MR 0601594 | Zbl 03779725
Search
Partner of
