Numerical ranges of nilpotent operators

Hwa Long Gau, Pei Yuan Wu

Research output: Contribution to journalArticlepeer-review

11 Scopus citations


For any operator A on a Hilbert space, let W (A), w (A) and w0 (A) denote its numerical range, numerical radius and the distance from the origin to the boundary of its numerical range, respectively. We prove that if An = 0, then w (A) ≤ (n - 1) w0 (A), and, moreover, if A attains its numerical radius, then the following are equivalent: (1) w (A) = (n - 1) w0 (A), (2) A is unitarily equivalent to an operator of the form aAn ⊕ A, where a is a scalar satisfying | a | = 2 w0 (A), An is the n-by-n matrixfenced((0, 1, ⋯, 1; 0, {triple dot, diagonal NW-SE}, ⋮; {triple dot, diagonal NW-SE}, 1; 0))andA is some other operator, and (3) W (A) = bW (An) for some scalar b.

Original languageEnglish
Pages (from-to)716-726
Number of pages11
JournalLinear Algebra and Its Applications
Issue number4
StatePublished - 1 Aug 2008


  • Nilpotent operator
  • Numerical radius
  • Numerical range


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