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).
Keywords
Mathematics Subject Classification (2000)
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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
T. Ando, On a pair of commutative contractions. Acta Sci. Math. (Szeged) 24, pp. 88–90 (1963)
C.A. Berger, A strange dilation theorem. Notices A.M.S. 12, p. 590 (1965)
R. Bouldin, The numerical range of a product II. J. Math, Anal. Appl. 33, pp. 212–219 (1971)
C.A. Berger and J.G. Stämpfli, Mapping theorems for the numerical radius. Amer. J. Math. 89, pp. 1047–1055 (1967)
V. Chkliar, Numerical radii of simple powers. Linear Algebra Appl. 265, pp. 119–121 (1997)
M.-T. Chien and B.-S. Tam, Circularity of the numerical range. Linear Algebra Appl. 201, pp. 113–133 (1994)
K.R. Davidson and J. Holbrook, Numerical radii for zero-one matrices. Michigan Math. J. 35, pp. 261–267 (1988)
F.M. Goodman, P. de la Harpe, and V.F.R. Jones, Coxeter Graphs and Towers of Algebras. Springer-Verlag (1989)
K.E. Gustafson and D.K.M. Rao, Numerical Range. Springer-Verlag (1997)
J. Holbrook, Multiplicative properties of the numerical radius in operator theory. J. Reine Angew. Math. 237, pp. 166–174 (1969)
J. Holbrook, Inequalities of von Neumann type for small matrices. Function Spaces (ed. K. Jarosz), pp. 189–193 (1992)
U. Haagerup and P. de la Harpe, The numerical radius of a nilpotent operator on a Hilbert space. Proceedings A.M.S. 115, pp. 371–379 (1992)
J. Holbrook and M. Omladič, Approximating commuting operators. Linear Algebra Appl. 327, pp. 131–149 (2001)
T. Kato, Some mapping theorems for the numerical range. Proc. Japan Acad. 41, pp. 652–655 (1965)
D.S. Keeler, L. Rodman, and I.M. Spitkovsky, The numerical range of3 × 3 matrices. Linear Algebra Appl. 252, pp. 115–139 (1997)
C.-K. Li and N.-S. Sze, Canonical forms, higher rank numerical ranges, totally isotropic subspaces, and matrix equations. Proc. Amer. Math. Soc. 136, pp. 3013–3023 (2008)
V. Müller, The numerical radius of a commuting product. Michigan Math. J. 35, pp. 255–260 (1988)
D.E.D. Marshall, An elementary proof of the Pick-Nevanlinna interpolation theorem. Michigan Math. J. 21, pp. 219–223 (1974)
M. Marcus and B.N. Shure, The numerical range of certain (0, 1)-matrices. Linear and Multilinear Algebra 7, no. 2, 111–120 (1979)
K. Okubo and T. Ando, Operator radii of commuting products. Proc. A.M.S. 56, pp. 203–210 (1976)
C. Pearcy, An elementary proof of the power inequality for the numerical radius. Michigan Math. J. 13, pp. 289–291 (1966)
http://en.wikipedia.org/wiki/Particle_swarm_optimization
http://en.wikipedia.org/wiki/Simulated_annealing
J.-P. Schoch, Theory vs Experiment in Matrix Analysis. PhD thesis, University of Guelph, Canada (2002)
Author information
Authors and Affiliations
Additional information
Dedicated to Leiba Rodman on the occasion of his 60th birthday
Communicated by V. Bolotnikov.
Rights and permissions
Copyright information
© 2010 Birkhäuser Verlag Basel/Switzerland
About this paper
Cite this paper
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
Download citation
DOI: https://doi.org/10.1007/978-3-0346-0158-0_14
Received:
Accepted:
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-0346-0157-3
Online ISBN: 978-3-0346-0158-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)