Numerical ranges of cube roots of the identity

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

Research output: Contribution to journalArticle

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 languageEnglish (US)
Pages (from-to)2639-2657
Number of pages19
JournalLinear Algebra and Its Applications
Volume435
Issue number11
DOIs
StatePublished - Dec 1 2011

Fingerprint

Cube root
Numerical Range
Hilbert spaces
Mathematical operators
Hilbert space
Set theory
Multiple Eigenvalues
Minimal polynomial
Polynomials
Threefolds
Bounded Linear Operator
Argand diagram
Symmetry
Subset
Operator

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

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

In: Linear Algebra and Its Applications, Vol. 435, No. 11, 01.12.2011, p. 2639-2657.

Research output: Contribution to journalArticle

Harris, TR, Mazzella, M, Patton, LJ, Renfrew, D & Spitkovsky, I 2011, 'Numerical ranges of cube roots of the identity', Linear Algebra and Its Applications, vol. 435, no. 11, pp. 2639-2657. https://doi.org/10.1016/j.laa.2011.03.020
Harris, Thomas Ryan ; Mazzella, Michael ; Patton, Linda J. ; Renfrew, David ; Spitkovsky, Ilya. / Numerical ranges of cube roots of the identity. In: Linear Algebra and Its Applications. 2011 ; Vol. 435, No. 11. pp. 2639-2657.
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