|Title: On a spectral version of Cartans theorem
Abstract: H.Cartan in 1931 proved the following theorem: every holomorphic self-map of a bounded domain (in the complex Euclidean space) that has a fixed point so that the derivative of the holomorphic map at the fixed point is identity has to be the identity map on the given bounded domain. Later, in 1970, S.Kobayashi generalized the above theorem to its strongest form; namely, one could replace bounded domains by Kobayashi hyperbolic complex manifolds in the above theorem. To illustrate to a reader unfamiliar with this topic, any planar domain that misses at least two distinct points in the complex plane is Kobayashi hyperbolic.
For a given domain Ω in the complex plane, we consider the matricial domains Sn(Ω) consisting of those n×n complex matrices whose spectrum is contained in Ω. The domains Sn(Ω) are not Kobayashi hyperbolic for any Ω and any n≥ 2. In this talk, we present results about holomorphic self-maps of Sn(Ω) in the spirit of Cartan’s Theorem. Our first result is: let Ψ be a holomorphic self-map of Sn(Ω) such that Ψ(A) = A and the derivative of Ψ at A is identity for some A ∈ Sn(Ω), then if A is either diagonalizable or non-derogatory then for most domains Ω, Ψ is spectrum-preserving on Sn(Ω). If time permits, we shall present another result; namely, for an arbitrary matrix A, Ψ is spectrum-preserving on a certain analytic subset of Sn(Ω).