Uniqueness theorems for analytic vector-valued functions

Using the Berezin transformation, we give a multidimensional analog of a uniqueness theorem of Nikolskii concerning distance functions and subspaces of a Hilbert space of analytic functions. Then, we establish some uniqueness properties connecting two analytic X-valued functions F and G that satisfy ∥F(z)∥ ≡ ∥G(z)∥ for all z ∈ Ω, where X is a Banach space and Ω is a connected domain in ℂn. The particular case where X = ℓnp and Ω = double-struck D sign = {z ∈ ℂ: |z| < 1} will lead us to the notion of flexible and inflexible functions. We give a complete description of these functions for p = +∞, n ∈ ℕ*, and for n = 2, 1 ≤ p ≤ + ∞. ©2000 Kluwer Academic/Plenum Publishers.