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On the spectral Nevanlinna-Pick problem

100%

Costara C.

Studia Mathematica

2005

tom 170

nr 1

23-55

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 method to construct analytic functions from the open unit disc of ℂ into Gₙ and to obtain some of the complex geodesics on Gₙ.

Nonlinear mappings preserving at least one eigenvalue

63%

Costara C., Repovš D.

Studia Mathematica

2010

tom 200

nr 1

79-89

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 $F(x) = uxu^{-1}$ or $F(x) = ux^{t}u^{-1}$ for all x, for some invertible n × n matrix u. We arrive at the same conclusion by supposing F to be of class 𝓒¹ on a domain in ℳₙ containing the null matrix, instead of Lipschitz. We also prove that if F is of class 𝓒¹ on a domain containing the null matrix satisfying F(0) = 0 and ρ(F(x) - F(y)) = ρ(x-y) for all x and y, where ρ(·) denotes the spectral radius, then there exists γ ∈ ℂ of modulus one such that either $γ^{-1}F$ or $γ^{-1}F̅$ is of the above form, where F̅ is the (complex) conjugate of F.

Ograniczanie wyników

2 Studia Mathematica

2 Costara C.

1 Repovš D.

1 2010

1 2005

Wyniki wyszukiwania

On the spectral Nevanlinna-Pick problem

Nonlinear mappings preserving at least one eigenvalue