Numerical ranges of cube roots of the identity

Thomas Ryan Harris; Michael Mazzella; Linda J. Patton; David Renfrew; Ilya M. Spitkovsky

doi:10.1016/j.laa.2011.03.020

Numerical ranges of cube roots of the identity

Thomas Ryan Harris, Michael Mazzella, Linda J. Patton, David Renfrew, Ilya M. Spitkovsky

Mathematics

Research output: Contribution to journal › Article › peer-review

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	https://doi.org/10.1016/j.laa.2011.03.020
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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10.1016/j.laa.2011.03.020

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AU - Harris, Thomas Ryan

AU - Mazzella, Michael

AU - Patton, Linda J.

AU - Renfrew, David

AU - Spitkovsky, Ilya M.

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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.

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Numerical ranges of cube roots of the identity

Abstract

Keywords

ASJC Scopus subject areas

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