On the automorphisms of Pick bodies

Abstract

We determine the biholomorphic automorphisms of the Pick body associated with finitely many distinct points in a Carathéodory hyperbolic domain. Every automorphism is the composition of a common disc automorphism applied to all coordinates and a coordinate permutation preserving the Pick body. The proof uses the quotient Banach algebra of interpolation data, its generalized von Neumann inequality, and the fact that its symmetric part consists precisely of the constant vectors. For nodes in a polydisc, we introduce minimal effective coordinate tuples and identify the admissible permutations with the automorphisms of a lower-dimensional polydisc that preserve the projected node set. This yields a direct-product description of the automorphism group, with one factor equal to the automorphism group of the disc and the other finite. We also give the resulting descriptions for nodes in the disc and the bidisc.