### Abstract

This paper provides a variety of sufficient conditions for the existence of a nonzero fixed point of a power-bounded linear operator defined on a real Banach space. In the case of power-bounded positive operators on a Banach lattice, among the conditions we provide are not being strongly stable along with commuting with a compact operator or being quasicompact. These results apply directly to Markov operators. In the case of an arbitrary power-bounded operator on a Hubert space, being uniformly asymptotically regular and not strongly stable guarantees the existence of a nonzero fixed point.

Original language | English (US) |
---|---|

Pages (from-to) | 1539-1551 |

Number of pages | 13 |

Journal | Proceedings of the American Mathematical Society |

Volume | 131 |

Issue number | 5 |

DOIs | |

State | Published - May 2003 |

### Fingerprint

### Keywords

- Asymptotic regularity
- Compact operators
- Contractions
- Fixed points
- Markov operators
- Strong stability

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Proceedings of the American Mathematical Society*,

*131*(5), 1539-1551. https://doi.org/10.1090/S0002-9939-02-06740-0

**Nonzero fixed points of power-bounded linear operators.** / Ok, Ahmet.

Research output: Contribution to journal › Article

*Proceedings of the American Mathematical Society*, vol. 131, no. 5, pp. 1539-1551. https://doi.org/10.1090/S0002-9939-02-06740-0

}

TY - JOUR

T1 - Nonzero fixed points of power-bounded linear operators

AU - Ok, Ahmet

PY - 2003/5

Y1 - 2003/5

N2 - This paper provides a variety of sufficient conditions for the existence of a nonzero fixed point of a power-bounded linear operator defined on a real Banach space. In the case of power-bounded positive operators on a Banach lattice, among the conditions we provide are not being strongly stable along with commuting with a compact operator or being quasicompact. These results apply directly to Markov operators. In the case of an arbitrary power-bounded operator on a Hubert space, being uniformly asymptotically regular and not strongly stable guarantees the existence of a nonzero fixed point.

AB - This paper provides a variety of sufficient conditions for the existence of a nonzero fixed point of a power-bounded linear operator defined on a real Banach space. In the case of power-bounded positive operators on a Banach lattice, among the conditions we provide are not being strongly stable along with commuting with a compact operator or being quasicompact. These results apply directly to Markov operators. In the case of an arbitrary power-bounded operator on a Hubert space, being uniformly asymptotically regular and not strongly stable guarantees the existence of a nonzero fixed point.

KW - Asymptotic regularity

KW - Compact operators

KW - Contractions

KW - Fixed points

KW - Markov operators

KW - Strong stability

UR - http://www.scopus.com/inward/record.url?scp=0037406122&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0037406122&partnerID=8YFLogxK

U2 - 10.1090/S0002-9939-02-06740-0

DO - 10.1090/S0002-9939-02-06740-0

M3 - Article

VL - 131

SP - 1539

EP - 1551

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 5

ER -