Numerical range of a pair of strictly upper triangular matrices

• Yan, Wen ^[1]
  1. [1] Tuskegee University

     Tuskegee University

     Estados Unidos

     
• Localización: Proyecciones: Journal of Mathematics, ISSN 0716-0917, ISSN-e 0717-6279, Vol. 30, Nº. 1, 2011, págs. 77-90
• Idioma: inglés
• DOI: 10.4067/S0716-09172011000100008
• Enlaces
  • Texto completo
• Resumen

  • Given two strictly upper triangular matrices X, Y ? Cm×m, we study the range WY (X) = {trnXn-1Y* : n ? N}, where N is the group of unit upper triangular matrices in Cm×m. We prove that it is either a point or the whole complex plane. We characterize when it is a point. We also obtain some convexity result for a similar range, where N is replaced by any ball of Ck(k = m(m - 1)/2) embedded in N , m = 4.

• Referencias bibliográficas
  • Citas [1] Y. H. Au-Yeung and N. K. Tsing, Some theorems on the generalized numerical range, Linear and Multilinear Algebra, 15, pp. 3—11,...
  • [2] C. Davis, The Toeplitz-Hausdorff theorem explained, Canad. Math. Bull., 14, pp. 245—246, (1971).
  • [3] W. S. Cheung and N. K. Tsing, The C-numerical range of matrices is star-shaped, Linear and Multilinear Algebra, 41, pp. 245—250, (1996).
  • [4] D. Z. Djokovic and T. Y. Tam, Some questions about semisimple Lie groups originating in matrix theory, Bull. Canad. Math. Soc., 46, pp....
  • [5] K. R. Gustafson and D. K. M. Rao, Numerical Range: The Field of Values of Linear Operators and Matrices, Universitext, SpringerVerlag,...
  • [6] V. G. Guti´errez and S. L. de Medrano, An extension of the ToeplitzHausdorff theorem, Bol. Soc. Mat. Mexicana (3), 9, pp. 273—278, (2003).
  • [7] P. R. Halmos, A Hilbert Space Problem Book, Springer-Verlag, New York, (1978).
  • [8] C. K. Li and T. Y. Tam, Numerical ranges arising from simple Lie algebras, J. Canad. Math. Soc., 52, pp. 141—171, (2000),
  • [9] R. Raghavendran, Toeplitz-Hausdorff theorem on numerical ranges, Proc. Amer. Math. Soc., 20, pp. 284—285, (1969).
  • [10] T. Y. Tam, An extension of a convexity theorem of the generalized numerical range associated with SO(2n + 1), Proc. Amer. Math. Soc.,...
  • [11] T. Y. Tam, Convexity of generalized numerical range associated with a compact Lie group, J. Austral. Math. Soc., 70, pp. 57—66, (2002).
  • [12] T. Y. Tam, On the shape of numerical range associated with Lie groups, Taiwanese J. Math., 5, pp. 497—506, (2001).
  • [13] N. K. Tsing, On the shape of the generalized numerical ranges, Linear and Multilinear Algebra, 10, pp. 173—182, (1981).
  • [14] R. Westwick, A theorem on numerical range, Linear and Multilinear Algebra, 2, pp. 311—315, (1975).