Numerical ranges of cube roots of the identity

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

Research output: Contribution to journalArticlepeer-review


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 languageEnglish (US)
Pages (from-to)2639-2657
Number of pages19
JournalLinear Algebra and Its Applications
Issue number11
StatePublished - Dec 1 2011


  • 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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