Noncircular elliptic discs as numerical ranges of nilpotent operators

Hwa Long Gau, Pei Yuan Wu

Research output: Contribution to journalArticlepeer-review

Abstract

We show that (1) if A is a nonzero quasinilpotent operator with ran A n closed for some n ≥ 1, then its numerical range W(A) contains 0 in its interior and has a differentiable boundary, and (2) a noncircular elliptic disc can be the numerical range of a nilpotent operator with nilpotency 3 on an infinite-dimensional separable space. (1) is a generalization of the known result for nonzero nilpotent operators, and (2) is in contrast to the finite-dimensional case, where the only elliptic discs which are the numerical ranges of nilpotent finite matrices are the circular ones centred at the origin.

Original languageEnglish
Pages (from-to)1225-1233
Number of pages9
JournalLinear and Multilinear Algebra
Volume60
Issue number11-12
DOIs
StatePublished - Nov 2012

Keywords

  • essential numerical range
  • nilpotent operator
  • numerical range
  • quasinilpotent operator

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