Abstract
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).
Several results are taken from Schoch’s doctoral thesis [Sch2002]. Holbrook’s work was supported in part by NSERC of Canada. The authors also thank David Kribs for helpful discussions. This work was partially supported by CFI, OIT, and other funding agencies.
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Dedicated to Leiba Rodman on the occasion of his 60th birthday
Communicated by V. Bolotnikov.
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Holbrook, J., Schoch, JP. (2010). Theory vs. Experiment: Multiplicative Inequalities for the Numerical Radius of Commuting Matrices. In: Topics in Operator Theory. Operator Theory: Advances and Applications, vol 202. Birkhäuser Basel. https://doi.org/10.1007/978-3-0346-0158-0_14
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DOI: https://doi.org/10.1007/978-3-0346-0158-0_14
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