### Abstract

Composition operators on the Hilbert Hardy space of the unit disk are considered. The shape of their numerical range is determined in the case when the symbol of the composition operator is a monomial or an inner function fixing 0. Several results on the numerical range of composition operators of arbitrary symbol are obtained. It is proved that 1 is an extreme boundary point if and only if 0 is a fixed point of the symbol. If 0 is not a fixed point of the symbol, 1 is shown to be interior to the numerical range. Some composition operators whose symbol fixes 0 and has infinity norm less than 1 have closed numerical ranges in the shape of a cone-like figure, i.e., a closed convex region with a corner at 1, 0 in its interior, and no other corners. Compact composition operators induced by a univalent symbol whose fixed point is not 0 have numerical ranges without corners, except possibly a corner at 0.

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

Pages (from-to) | 61-74 |

Number of pages | 14 |

Journal | Linear Algebra and Its Applications |

Volume | 331 |

Issue number | 1-3 |

DOIs | |

State | Published - Jul 1 2001 |

### Fingerprint

### Keywords

- 47A12
- 47B38
- Composition operator
- Hardy space
- Numerical range

### ASJC Scopus subject areas

- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics

### Cite this

**Numerical ranges of composition operators.** / Matache, Valentin.

Research output: Contribution to journal › Article

*Linear Algebra and Its Applications*, vol. 331, no. 1-3, pp. 61-74. https://doi.org/10.1016/S0024-3795(01)00262-2

}

TY - JOUR

T1 - Numerical ranges of composition operators

AU - Matache, Valentin

PY - 2001/7/1

Y1 - 2001/7/1

N2 - Composition operators on the Hilbert Hardy space of the unit disk are considered. The shape of their numerical range is determined in the case when the symbol of the composition operator is a monomial or an inner function fixing 0. Several results on the numerical range of composition operators of arbitrary symbol are obtained. It is proved that 1 is an extreme boundary point if and only if 0 is a fixed point of the symbol. If 0 is not a fixed point of the symbol, 1 is shown to be interior to the numerical range. Some composition operators whose symbol fixes 0 and has infinity norm less than 1 have closed numerical ranges in the shape of a cone-like figure, i.e., a closed convex region with a corner at 1, 0 in its interior, and no other corners. Compact composition operators induced by a univalent symbol whose fixed point is not 0 have numerical ranges without corners, except possibly a corner at 0.

AB - Composition operators on the Hilbert Hardy space of the unit disk are considered. The shape of their numerical range is determined in the case when the symbol of the composition operator is a monomial or an inner function fixing 0. Several results on the numerical range of composition operators of arbitrary symbol are obtained. It is proved that 1 is an extreme boundary point if and only if 0 is a fixed point of the symbol. If 0 is not a fixed point of the symbol, 1 is shown to be interior to the numerical range. Some composition operators whose symbol fixes 0 and has infinity norm less than 1 have closed numerical ranges in the shape of a cone-like figure, i.e., a closed convex region with a corner at 1, 0 in its interior, and no other corners. Compact composition operators induced by a univalent symbol whose fixed point is not 0 have numerical ranges without corners, except possibly a corner at 0.

KW - 47A12

KW - 47B38

KW - Composition operator

KW - Hardy space

KW - Numerical range

UR - http://www.scopus.com/inward/record.url?scp=0035400206&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0035400206&partnerID=8YFLogxK

U2 - 10.1016/S0024-3795(01)00262-2

DO - 10.1016/S0024-3795(01)00262-2

M3 - Article

AN - SCOPUS:0035400206

VL - 331

SP - 61

EP - 74

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - 1-3

ER -