Toeplitz-Hausdorff systems Academic Article uri icon

abstract

  • The numerical range of an n × n matrix T is the image of T under a certain set of linear functionals-a set that comprises the extreme points among the states (i.e., norm-one, positive linear functionals) on the n × n matrices-and is convex, by the Toeplitz-Hausdorff theorem. One can view this convexity as a consequence of T's numerical range being equal to a manifestly convex set, the image of T under all states. Taking this view leads us to ask whether a similar result holds when we replace the n × n matrices by a finite dimensional Banach space {A figure is presented}, the states by a closed, convex subset Σ of {A figure is presented}*, and the extreme states by the extreme points of Σ. When it does, we call the pair ({A figure is presented}, Σ) a Toeplitz-Hausdorff system. In this paper, we show that if {A figure is presented} is what we term a nullifying subspace of the n×n matrices, and if Σ is the closed unit ball in {A figure is presented}*, then ({A figure is presented}, Σ) is a Toeplitz-Hausdorff system. (Both the upper and lower triangular matrices form nullifying subspaces.). © 1984.

author list (cited authors)

  • Narcowich, F. J., & Ward, J. D.

citation count

  • 0

publication date

  • November 1984