### 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) |
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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 |

### Keywords

- Algebraic operator
- Numerical range
- Threefold symmetry

### ASJC Scopus subject areas

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

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## Cite this

Harris, T. R., Mazzella, M., Patton, L. J., Renfrew, D., & Spitkovsky, I. M. (2011). Numerical ranges of cube roots of the identity.

*Linear Algebra and Its Applications*,*435*(11), 2639-2657. https://doi.org/10.1016/j.laa.2011.03.020