Theory vs. Experiment: Multiplicative Inequalities for the Numerical Radius of Commuting Matrices

  • John Holbrook
  • Jean-Pierre Schoch
Part of the Operator Theory: Advances and Applications book series (OT, volume 202)


Under what conditions does the inequality w(TS) ≤ w(T)∥S∥, or the stronger w(TS) ≤ w(T)w(S), hold? Here w(T) denotes the numerical radius max{∣(Tu,u)∣: ∥u∥ = 1} of the matrix T and ∥S∥ is the operator norm; we assume that T and S are commuting n×n matrices. The questions posed above have a long history in matrix analysis and this paper provides new information, combining theoretical and experimental approaches. We study a class of matrices with simple structure to reveal a variety of new counterexamples to the first inequality. By means of carefully designed computer experiments we show that the first inequality may fail even for 3×3 matrices. We also obtain bounds on the constant that must be inserted in the second inequality when the matrices are 3 × 3. Among other results, we obtain new instances of the phenomenon discovered by Chkliar: for certain contractions C we may have w(C m+1) < w(C m ).


Numerical radius matrix norm inequalities 

Mathematics Subject Classification (2000)

15A60 15A04 47A49 


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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2010

Authors and Affiliations

  • John Holbrook
    • 1
  • Jean-Pierre Schoch
    • 2
  1. 1.Dept. of Mathematics and StatisticsUniversity of GuelphGuelphCanada
  2. 2.ArkellOntarioCanada

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