Proper Holomorphic Mappings of the Spectral Unit Ball

We prove an Alexander type theorem for the spectral unit ball Ωn showing that there are no non-trivial proper holomorphic mappings in Ωn, n ≥ 2. Let Mn denote the space of n× n complex matrices. In order to avoid some trivialities and ambiguities we assume in the whole paper that n ≥ 2. Let ρ(A) := max{|λ| : λ ∈ Spec(A)} be the spectral radius of A ∈ Mn. Denote also by Spec(A) := {λ ∈ C : det(A − λIn) = 0} the spectrum of A ∈ Mn, where the eigenvalues are counted with multiplicities (In denotes the identity matrix). We also denote the spectral unit ball by Ωn := {A ∈ Mn : ρ(A) < 1}. Note that Ωn is an unbounded pseudoconvex balanced domain in C 2 with the continuous Minkowski functional equal to ρ. For A ∈ Mn denote PA(λ) := det(λIn − A) = λ + ∑n j=1(−1)σj(A)λ , A ∈ Mn. Denote also σ := (σ1, . . . , σn). We put Gn := σ(Ωn). The domain Gn is called the symmetrized polydisc. Note that σ ∈ O(Mn,Gn). Denote also Jn := πn({(ζ1, . . . , ζn) : ζj = ζk for some j = k}), where πn,j(ζ1, . . . , ζn) := ∑ 1≤k1<...<kj≤n ζk1 · . . . · ζkj , ζl ∈ D, l = 1, . . . , n (D denotes the unit disc in C). Note that Gn \Jn is a domain and Gn \Jn is dense in Gn. Note that Ωn = ⋃ z∈Gn Tz, where Tz := {A ∈ Ωn : σ(A) = z}, z ∈ C . The sets Tz, z ∈ C, are pairwise disjoint analytic sets. Note that if the matrix A ∈ Tz is nonderogatory, then A is a regular point of Tz (recall that in such a case rankσ′(A) = n); it is the largest possible number. For a definition and basic properties of nonderogatory matrices see [Nik-Tho-Zwo 2007] and the references therein. One of the possible definitions of a non-derogatory matrix is that different blocks in the Jordan normal form correspond to different eigenvalues (or equivalently all eigenspaces are one-dimensional). We shall deliver some properties of the sets Tz (see Lemma 5, Lemma 6 and Corollary 7). It is also simple to see that T0 is a cone which contains at least n − n+ 1 linearly independent vectors: for instance the ones consisting of one 1 not lying on the diagonal (and with other entries equal to 0) and the matrix Received by the editors April 5, 2007. 2000 Mathematics Subject Classification. Primary 32H35; Secondary 15A18, 32C25, 47N99.