Holomorphic maps preserving parts of the local spectrum

Let x ₀ be a nonzero vector in ({mathbb{C}^{n}}) , and let ({Usubseteq mathcal{M}_{n}}) be a domain containing the zero matrix. We prove that if φ is a holomorphic map from U into ({mathcal{M}_{n}}) such that the local spectrum of T ∈ U at x ₀ and the local spectrum of φ(T) at x ₀ have always a common value, then T and φ(T) have always the same spectrum, and they have the same local spectrum at x ₀ a.e. with respect to the Lebesgue measure on U. If ({varphi colon Urightarrow mathcal{M}_{n}}) is holomorphic with φ(0) = 0 such that the local spectral radius of T at x ₀ equals the local spectral radius of φ(T) at x ₀ for all T ∈ U, there exists ({xi in mathbb{C}}) of modulus one such that ξT and φ(T) have the same spectrum for all T in U. We also prove that if for all T ∈ U the local spectral radius of φ(T) coincides with the local spectral radius of T at each vector x, there exists ({xi in mathbb{C}}) of modulus one such that φ(T) = ξT on U.