Geodesic
On invariant subspaces for polynomially bounded operators
Czechoslovak Mathematical Journal, Tome 67 (2017) no. 1, pp. 1-9 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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).
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).
DOI : 10.21136/CMJ.2017.0459-14
Classification : 47A15
Keywords: polynomially bounded operator; invariant subspace 
@article{10_21136_CMJ_2017_0459_14,
     author = {Liu, Junfeng},
     title = {On invariant subspaces for polynomially bounded operators},
     journal = {Czechoslovak Mathematical Journal},
     pages = {1--9},
     year = {2017},
     volume = {67},
     number = {1},
     doi = {10.21136/CMJ.2017.0459-14},
     mrnumber = {3632994},
     zbl = {06738500},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2017.0459-14/}
}
TY  - JOUR
AU  - Liu, Junfeng
TI  - On invariant subspaces for polynomially bounded operators
JO  - Czechoslovak Mathematical Journal
PY  - 2017
SP  - 1
EP  - 9
VL  - 67
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2017.0459-14/
DO  - 10.21136/CMJ.2017.0459-14
LA  - en
ID  - 10_21136_CMJ_2017_0459_14
ER  - 
%0 Journal Article
%A Liu, Junfeng
%T On invariant subspaces for polynomially bounded operators
%J Czechoslovak Mathematical Journal
%D 2017
%P 1-9
%V 67
%N 1
%U http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2017.0459-14/
%R 10.21136/CMJ.2017.0459-14
%G en
%F 10_21136_CMJ_2017_0459_14
Liu, Junfeng. On invariant subspaces for polynomially bounded operators. Czechoslovak Mathematical Journal, Tome 67 (2017) no. 1, pp. 1-9. doi: 10.21136/CMJ.2017.0459-14

[1] Ambrozie, C., Müller, V.: Invariant subspaces for polynomially bounded operators. J. Funct. Anal. 213 (2004), 321-345. | DOI | MR | JFM

[2] Beauzamy, B.: Introduction to Operator Theory and Invariant Subspaces. North-Holland Mathematical Library 42, North-Holland, Amsterdam (1988). | MR | JFM

[3] Brown, S. W., Chevreau, B., Pearcy, C.: On the structure of contraction operators. II. J. Funct. Anal. 76 (1988), 30-55. | DOI | MR | JFM

[4] Chalendar, I., Partington, J. R.: Modern Approaches to the Invariant-Subspace Problem. Cambridge Tracts in Mathematics 188, Cambridge University Press, Cambridge (2011). | MR | JFM

[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

[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 | JFM

[7] Lomonosov, V.: An extension of Burnside's theorem to infinite-dimensional spaces. Isr. J. Math. 75 (1991), 329-339. | DOI | MR | JFM

[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 | MR | JFM

[9] Rudin, W.: Function Theory in the Unit Ball of $C^n$. Grundlehren der mathematischen Wissenschaften 241, Springer, Berlin (1980). | DOI | MR | JFM

Cité par Sources :