Effective determination of all the Hahn--Banach extensions of some linear and continuous functionals.
On the spectral Nevanlinna-Pick problem
We give several characterizations of the symmetrized n-disc Gₙ which generalize to the case n ≥ 3 the characterizations of the symmetrized bidisc that were used in order to solve the two-point spectral Nevanlinna-Pick problem in ℳ ₂(ℂ). Using these characterizations of the symmetrized n-disc, which give necessary and sufficient conditions for an element to belong to Gₙ, we obtain necessary conditions of interpolation for the general spectral Nevanlinna-Pick problem. They also allow us to give a...
Nonlinear mappings preserving at least one eigenvalue
We prove that if F is a Lipschitz map from the set of all complex n × n matrices into itself with F(0) = 0 such that given any x and y we know that F(x) - F(y) and x-y have at least one common eigenvalue, then either or for all x, for some invertible n × n matrix u. We arrive at the same conclusion by supposing F to be of class ¹ on a domain in ℳₙ containing the null matrix, instead of Lipschitz. We also prove that if F is of class ¹ on a domain containing the null matrix satisfying F(0) = 0...
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