A uniqueness result concerning Schur ideals

Abstract

A subset of the set of all positive semi-definite matrices of a given size which is invariant under Schur (componentwise) multiplication by an arbitrary positive semi-definite matrix is said to be a Schur ideal. A subset of k k -dimensional complex space is said to be h y p e r c o n v e x hyperconvex if it arises as the set of possible values ( w 1 , … , w k ) = ( f ( α 1 ) , … , f ( α k ) ) (w_{1}, \dots , w_{k}) = (f(\alpha _{1}), \dots , f(\alpha _{k})) arising from restricting contractive elements f f from some uniform algebra A A to a finite set { α 1 , … , α k } \{ \alpha _{1}, \dots , \alpha _{k} \} in the domain. When the uniform algebra is the disk algebra, the hyperconvex set is said to be a Pick body. Motivated by the classical Pick interpolation theorem, Paulsen has introduced a natural notion of duality between Schur ideals and hyperconvex sets. By using some recently developed results in operator algebras (matricial Schur ideals), we show that each Pick body has a unique affiliated Schur ideal.