On invariant subspaces for polynomially bounded operators

Junfeng Liu

Czechoslovak Mathematical Journal (2017)

  • Volume: 67, Issue: 1, page 1-9
  • ISSN: 0011-4642

Abstract

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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).

How to cite

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Liu, Junfeng. "On invariant subspaces for polynomially bounded operators." Czechoslovak Mathematical Journal 67.1 (2017): 1-9. <http://eudml.org/doc/287930>.

@article{Liu2017,
abstract = {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).},
author = {Liu, Junfeng},
journal = {Czechoslovak Mathematical Journal},
keywords = {polynomially bounded operator; invariant subspace},
language = {eng},
number = {1},
pages = {1-9},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On invariant subspaces for polynomially bounded operators},
url = {http://eudml.org/doc/287930},
volume = {67},
year = {2017},
}

TY - JOUR
AU - Liu, Junfeng
TI - On invariant subspaces for polynomially bounded operators
JO - Czechoslovak Mathematical Journal
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 67
IS - 1
SP - 1
EP - 9
AB - 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).
LA - eng
KW - polynomially bounded operator; invariant subspace
UR - http://eudml.org/doc/287930
ER -

References

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  1. Ambrozie, C., Müller, V., 10.1016/j.jfa.2003.12.004, J. Funct. Anal. 213 (2004), 321-345. (2004) Zbl1056.47006MR2078629DOI10.1016/j.jfa.2003.12.004
  2. Beauzamy, B., Introduction to Operator Theory and Invariant Subspaces, North-Holland Mathematical Library 42, North-Holland, Amsterdam (1988). (1988) Zbl0663.47002MR0967989
  3. Brown, S. W., Chevreau, B., Pearcy, C., 10.1016/0022-1236(88)90047-X, J. Funct. Anal. 76 (1988), 30-55. (1988) Zbl0641.47013MR0923043DOI10.1016/0022-1236(88)90047-X
  4. Chalendar, I., Partington, J. R., Modern Approaches to the Invariant-Subspace Problem, Cambridge Tracts in Mathematics 188, Cambridge University Press, Cambridge (2011). (2011) Zbl1231.47005MR2841051
  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). (2001) MR2702823
  6. Laursen, K. B., Neumann, M. M., An Introduction to Local Spectral Theory, London Mathematical Society Monographs. New Series 20, Clarendon Press, Oxford (2000). (2000) Zbl0957.47004MR1747914
  7. Lomonosov, V., 10.1007/BF02776031, Isr. J. Math. 75 (1991), 329-339. (1991) Zbl0777.47005MR1164597DOI10.1007/BF02776031
  8. Pisier, G., 10.1090/S0894-0347-97-00227-0, J. Am. Math. Soc. 10 (1997), 351-369. (1997) Zbl0869.47014MR1415321DOI10.1090/S0894-0347-97-00227-0
  9. Rudin, W., 10.1007/978-3-540-68276-9, Grundlehren der mathematischen Wissenschaften 241, Springer, Berlin (1980). (1980) Zbl03779725MR0601594DOI10.1007/978-3-540-68276-9

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