Hostname: page-component-84b7d79bbc-5lx2p Total loading time: 0 Render date: 2024-07-30T13:41:20.841Z Has data issue: false hasContentIssue false
  • English
  • Français

H-joint numerical ranges

Published online by Cambridge University Press:  17 April 2009

Chi-Kwong Li  and
Leiba Rodman
Chi-Kwong Li
Affiliation:
Department of Mathematics, College of William and Mary, Williamsburg, VA 23187–8795, United States of America e-mail: ckli@wm.edu, lxrodm@math.wm.edu
Leiba Rodman
Affiliation:
Department of Mathematics, College of William and Mary, Williamsburg, VA 23187–8795, United States of America e-mail: ckli@wm.edu, lxrodm@math.wm.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The notion of the joint numerical range of several linear operators with respect to a sesquilinear form is introduced. Geometrical properties of the joint numerical range are studied, in particular, convexity and angle points, in connection with the algebraic properties of the operators. The main focus is on the finite dimensional case.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 66 , Issue 1 , August 2002 , pp. 105 - 117
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Au-Yeung, Y.H. and Tsing, N.K., ‘An extension of the Hausdorff-Toeplitz theorem on the numerical range’, Proc. Amer. Math. Soc. 89 (1983), 215–218.CrossRefGoogle Scholar
[2]Binding, P. and Li, C.K., ‘Joint numerical ranges of Hermitian matrices and simultaneous diagonalization’, Linear Algebra Appl. 151 (1991), 157–168.CrossRefGoogle Scholar
[3]Donoghue, W.F. Jr., ‘On the numerical range of a bounded operator’, Michigan Math. J. 4 (1957), 261–263.CrossRefGoogle Scholar
[4]Fan, M.K.H. and Tits, A.L., ‘m-form numerical range and the computation of the structured singular value’, IEEE Trans. Automat. Control 33 (1988), 284–289.CrossRefGoogle Scholar
[5]Li, C.K., ‘A simple proof of the elliptical range theorem’, Proc. Amer. Math. Soc. 124 (1996), 1985–1986.CrossRefGoogle Scholar
[6]Li, C.K., ‘The c-spectral, c-radial and c-convex matrices’, Linear and Multilinear Algebra 20 (1986), 5–15.CrossRefGoogle Scholar
[7]Li, C.K. and Poon, Y.T., ‘Convexity of the joint numerical range’, SIAM J. Matrix Anal. Appl. 21 (2000), 668–678.CrossRefGoogle Scholar
[8]Li, C.K. and Rodman, L., ‘Remarks on numerical ranges of operators in spaces with an indefinite metric’, Proc. Amer. Math. Soc. 126 (1998), 973–982.CrossRefGoogle Scholar
[9]Li, C.K. and Tam, T.Y., ‘Numerical range arising from simple Lie algebras’, Canad. J. Math. 52 (2000), 141–171.CrossRefGoogle Scholar
[10]Li, C.K., Tsing, N.K. and Uhlig, F., ‘Numerical range of an operator on an indefinite inner product space’, Electron. J. Linear Algebra 1 (1996), 1–17.CrossRefGoogle Scholar
[11]Marcus, M. and Filippenko, I., ‘Nondifferentiable boundary points of higher numerical range’, Linear Algebra Appl. 21 (1978), 217–232.CrossRefGoogle Scholar
[12]Poon, Y.T., ‘On the convex hull of the multiform numerical range’, Linear and Multilinear Algebra 37 (1994), 221–223.CrossRefGoogle Scholar
[13]Poon, Y.T., ‘Generalized numerical ranges, joint positive definiteness and multiple eigen-values’, Proc. Amer. Math. Soc. 125 (1997), 1625–1634.CrossRefGoogle Scholar
[14]Wang, J.-C., Fan, M.K.H. and Tits, A.L., ‘Structured singular value and geometry of the m-form numerical range’, in Linear circuits, systems and signal processing: theory and application (Phoenix, AZ, 1987), (North-Holland, Amsterdam, 1988), pp. 609–615.Google Scholar