A Local Form for the Automorphisms of the Spectral Unit Ball

If F is an automorphism of Ωn, the n -dimensional spectral unit ball, we show that, in a neighborhood of any cyclic matrix of Ωn, the map F can be written as conjugation by a holomorphically varying non singular matrix. This provides a shorter proof of a theorem of J. Rostand, with a slightly stronger result. 1. Background Let Mn be the set of all n × n complex matrices. For A ∈ Mn denote by sp(A) the spectrum of A. The spectral ball Ωn is the set Ωn := {A ∈ Mn : ∀λ ∈ sp(A), |λ| < 1}. Let F be an automorphism of Ωn, that is to say, a biholomorphic map of the spectral ball into itself. Ransford and White [6] proved that, by composing with a natural lifting of a Möbius map of the disk, one could reduce oneself to the case where F (0) = 0, and that in that case the linear map F (0) was a linear automorphism of Ωn, so that by composing with its inverse, one is reduced to the case F (0) = 0, F (0) = I (the identity map). We then say that the automorphism if normalized. Ransford and White [6] proved that such automorphisms preserve the spectrum of matrices. We say that two matrices X, Y are conjugate if there exists Q ∈ M n such that X = QY Q. Baribeau and Ransford [1] (see also [2] for a more elementary proof) proved that every spectrum-preserving C-diffeomorphism of an open subset of Mn, and thus every normalized automorphism of the spectral ball is a pointwise conjugation: (1) F (X) = Q(X)XQ(X). 2000 Mathematics Subject Classification. 32H02, 32A07, 15A21.