### Abstract

In this paper, we give a brief review concerning diskcyclic operators and then we provide some further characterizations of diskcyclic operators on separable Hilbert spaces. In particular, we show that if (Formula presented.) has a disk orbit under (Formula presented.) that is somewhere dense in (Formula presented.) , then the disk orbit of (Formula presented.) under (Formula presented.) need not be everywhere dense in (Formula presented.). We also show that the inverse and the adjoint of a diskcyclic operator need not be diskcyclic. Moreover, we establish another diskcyclicity criterion and use it to find a necessary and sufficient condition for unilateral backward shifts that are diskcyclic operators. We show that a diskcyclic operator exists on a Hilbert space (Formula presented.) over the field of complex numbers if and only if (Formula presented.) or (Formula presented.) . Finally, we give a sufficient condition for the somewhere density disk orbit to be everywhere dense.

Original language | English |
---|---|

Pages (from-to) | 723-739 |

Number of pages | 17 |

Journal | Bulletin of the Malaysian Mathematical Sciences Society |

Volume | 39 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1 Apr 2016 |

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### Keywords

- Diskcyclic operators
- Hypercyclic operators
- Supercyclic operators
- Weighted shift operators

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Bulletin of the Malaysian Mathematical Sciences Society*,

*39*(2), 723-739. https://doi.org/10.1007/s40840-015-0137-x

**A Review of Some Works in the Theory of Diskcyclic Operators.** / Bamerni, Nareen; Kılıçman, Adem; Md. Noorani, Mohd. Salmi.

Research output: Contribution to journal › Article

*Bulletin of the Malaysian Mathematical Sciences Society*, vol. 39, no. 2, pp. 723-739. https://doi.org/10.1007/s40840-015-0137-x

}

TY - JOUR

T1 - A Review of Some Works in the Theory of Diskcyclic Operators

AU - Bamerni, Nareen

AU - Kılıçman, Adem

AU - Md. Noorani, Mohd. Salmi

PY - 2016/4/1

Y1 - 2016/4/1

N2 - In this paper, we give a brief review concerning diskcyclic operators and then we provide some further characterizations of diskcyclic operators on separable Hilbert spaces. In particular, we show that if (Formula presented.) has a disk orbit under (Formula presented.) that is somewhere dense in (Formula presented.) , then the disk orbit of (Formula presented.) under (Formula presented.) need not be everywhere dense in (Formula presented.). We also show that the inverse and the adjoint of a diskcyclic operator need not be diskcyclic. Moreover, we establish another diskcyclicity criterion and use it to find a necessary and sufficient condition for unilateral backward shifts that are diskcyclic operators. We show that a diskcyclic operator exists on a Hilbert space (Formula presented.) over the field of complex numbers if and only if (Formula presented.) or (Formula presented.) . Finally, we give a sufficient condition for the somewhere density disk orbit to be everywhere dense.

AB - In this paper, we give a brief review concerning diskcyclic operators and then we provide some further characterizations of diskcyclic operators on separable Hilbert spaces. In particular, we show that if (Formula presented.) has a disk orbit under (Formula presented.) that is somewhere dense in (Formula presented.) , then the disk orbit of (Formula presented.) under (Formula presented.) need not be everywhere dense in (Formula presented.). We also show that the inverse and the adjoint of a diskcyclic operator need not be diskcyclic. Moreover, we establish another diskcyclicity criterion and use it to find a necessary and sufficient condition for unilateral backward shifts that are diskcyclic operators. We show that a diskcyclic operator exists on a Hilbert space (Formula presented.) over the field of complex numbers if and only if (Formula presented.) or (Formula presented.) . Finally, we give a sufficient condition for the somewhere density disk orbit to be everywhere dense.

KW - Diskcyclic operators

KW - Hypercyclic operators

KW - Supercyclic operators

KW - Weighted shift operators

UR - http://www.scopus.com/inward/record.url?scp=84960419398&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84960419398&partnerID=8YFLogxK

U2 - 10.1007/s40840-015-0137-x

DO - 10.1007/s40840-015-0137-x

M3 - Article

VL - 39

SP - 723

EP - 739

JO - Bulletin of the Malaysian Mathematical Sciences Society

JF - Bulletin of the Malaysian Mathematical Sciences Society

SN - 0126-6705

IS - 2

ER -