### Abstract

The numerical range of a bounded linear operator T on a Hilbert space H is defined to be the subset W(T)={〈Tv,v〉:v∈H,∥v∥=1} of the complex plane. For operators on a finite-dimensional Hilbert space, it is known that if W(T) is a circular disk then the center of the disk must be a multiple eigenvalue of T. In particular, if T has minimal polynomial ^{z3}-1, then W(T) cannot be a circular disk. In this paper we show that this is no longer the case when H is infinite dimensional. The collection of 3×3 matrices with three-fold symmetry about the origin are also classified.

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

Pages (from-to) | 2639-2657 |

Number of pages | 19 |

Journal | Linear Algebra and Its Applications |

Volume | 435 |

Issue number | 11 |

DOIs | |

State | Published - Dec 1 2011 |

### Fingerprint

### Keywords

- Algebraic operator
- Numerical range
- Threefold symmetry

### ASJC Scopus subject areas

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

### Cite this

*Linear Algebra and Its Applications*,

*435*(11), 2639-2657. https://doi.org/10.1016/j.laa.2011.03.020

**Numerical ranges of cube roots of the identity.** / Harris, Thomas Ryan; Mazzella, Michael; Patton, Linda J.; Renfrew, David; Spitkovsky, Ilya.

Research output: Contribution to journal › Article

*Linear Algebra and Its Applications*, vol. 435, no. 11, pp. 2639-2657. https://doi.org/10.1016/j.laa.2011.03.020

}

TY - JOUR

T1 - Numerical ranges of cube roots of the identity

AU - Harris, Thomas Ryan

AU - Mazzella, Michael

AU - Patton, Linda J.

AU - Renfrew, David

AU - Spitkovsky, Ilya

PY - 2011/12/1

Y1 - 2011/12/1

N2 - The numerical range of a bounded linear operator T on a Hilbert space H is defined to be the subset W(T)={〈Tv,v〉:v∈H,∥v∥=1} of the complex plane. For operators on a finite-dimensional Hilbert space, it is known that if W(T) is a circular disk then the center of the disk must be a multiple eigenvalue of T. In particular, if T has minimal polynomial z3-1, then W(T) cannot be a circular disk. In this paper we show that this is no longer the case when H is infinite dimensional. The collection of 3×3 matrices with three-fold symmetry about the origin are also classified.

AB - The numerical range of a bounded linear operator T on a Hilbert space H is defined to be the subset W(T)={〈Tv,v〉:v∈H,∥v∥=1} of the complex plane. For operators on a finite-dimensional Hilbert space, it is known that if W(T) is a circular disk then the center of the disk must be a multiple eigenvalue of T. In particular, if T has minimal polynomial z3-1, then W(T) cannot be a circular disk. In this paper we show that this is no longer the case when H is infinite dimensional. The collection of 3×3 matrices with three-fold symmetry about the origin are also classified.

KW - Algebraic operator

KW - Numerical range

KW - Threefold symmetry

UR - http://www.scopus.com/inward/record.url?scp=79960838570&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79960838570&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2011.03.020

DO - 10.1016/j.laa.2011.03.020

M3 - Article

VL - 435

SP - 2639

EP - 2657

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - 11

ER -