Equality of three numerical radius inequalities

Hwa Long Gau, Pei Yuan Wu

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


For an n-by-n matrix A, let w(A) and ‖A‖ denote its numerical radius and operator norm, respectively. The following three inequalities, each a strengthening of w(A)≤‖A‖, are known to hold: w(A)2≤(‖A‖2+w(A2))/2, w(A)≤(‖A‖+‖A21/2)/2, and w(A)≤(‖A‖+w(Δt(A)))/2 (0≤t≤1), where Δt(A) is the generalized Aluthge transform of A. In this paper, we derive necessary and sufficient conditions in terms of the operator structure of A for which the inequalities become equalities.

Original languageEnglish
Pages (from-to)51-67
Number of pages17
JournalLinear Algebra and Its Applications
StatePublished - 1 Oct 2018


  • Generalized Aluthge transform
  • Numerical radius
  • Numerical range
  • Operator norm


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