Some remarks on distributional chaos for bounded linear operators

Lvlin LUO; Bingzhe HOU

Article Details

Some remarks on distributional chaos for bounded linear operators

Authors:

Lvlin LUO | Bingzhe HOU

Publisher:

TUBITAK

Journal:

Turkish Journal of Mathematics 2015 Vol.39 Issue 2, pp.251-258

Document Type:

Article

Publication Date:

3/2015

Document Language:

Turkish

Document ID (Org No.):

2000747209

Frequency:

Bimonthly

eISSN:

1303-6149

pISSN:

1300-0098

Keywords:

Abstract:

 

The aim of this paper is to study distributional chaos for bounded linear operators. We show that distributional chaos of type k \in {1,2} is an invariant of topological conjugacy between two bounded linear operators. We give a necessary condition for distributional chaos of type 2 where it is possible to distinguish distributional chaos and Li--Yorke chaos. Following this condition, we compare distributional chaos with other well-studied notions of chaos for backward weighted shift operators and give an alternative proof to the one where strong mixing does not imply distributional chaos of type 2 (Martínez-Giménez F, Oprocha P, Peris A. Distributional chaos for operators with full scrambled sets. Math Z 2013; 274: 603--612.). Moreover, we also prove that there exists an invertible bilateral forward weighted shift operator such that it is DC1 but its inverse is not DC2.

 

The aim of this paper is to study distributional chaos for bounded linear operators. We show that distributional chaos of type k \in {1,2} is an invariant of topological conjugacy between two bounded linear operators. We give a necessary condition for distributional chaos of type 2 where it is possible to distinguish distributional chaos and Li--Yorke chaos. Following this condition, we compare distributional chaos with other well-studied notions of chaos for backward weighted shift operators and give an alternative proof to the one where strong mixing does not imply distributional chaos of type 2 (Martínez-Giménez F, Oprocha P, Peris A. Distributional chaos for operators with full scrambled sets. Math Z 2013; 274: 603--612.). Moreover, we also prove that there exists an invertible bilateral forward weighted shift operator such that it is DC1 but its inverse is not DC2.

Article Details

Some remarks on distributional chaos for bounded linear operators

Authors:

Lvlin LUO | Bingzhe HOU

Publisher:

TUBITAK

Journal:

Turkish Journal of Mathematics 2015 Vol.39 Issue 2, pp.251-258

Document Type:

Article

Publication Date:

3/2015

Document Language:

Turkish

Document ID (Org No.):

2000747209

Frequency:

Bimonthly

eISSN:

1303-6149

pISSN:

1300-0098

Distributional chaos

operators

weighted shifts

topological conjugacy

strong mixing

Issues

Article Details

Some remarks on distributional chaos for bounded linear operators

Authors: Lvlin LUO | Bingzhe HOU

Publisher: TUBITAK

Journal: Turkish Journal of Mathematics 2015 Vol.39 Issue 2, pp.251-258

Document Type: Article

Publication Date: 3/2015

Document Language: Turkish

Document ID (Org No.): 2000747209

Frequency: Bimonthly

eISSN: 1303-6149

pISSN: 1300-0098

Distributional chaos

operators

weighted shifts

topological conjugacy

strong mixing

Issues

Authors:

Lvlin LUO | Bingzhe HOU

Publisher:

TUBITAK

Journal:

Turkish Journal of Mathematics 2015 Vol.39 Issue 2, pp.251-258

Document Type:

Article

Publication Date:

3/2015

Document Language:

Turkish

Document ID (Org No.):

2000747209

Frequency:

Bimonthly

eISSN:

1303-6149

pISSN:

1300-0098