An Improved Julia-caratheodory Theorem for Schur-agler Mappings of the Unit Ball

We adapt Sarason’s proof of the Julia-Caratheodory theorem to the class of Schur-Agler mappings of the unit ball, obtaining a strengthened form of this theorem. In particular those quantities which appear in the classical theorem and depend only on the component of the mapping in the complex normal direction have K-limits (not just restricted K-limits) at the boundary. Let B denote the open unit ball in n-dimensional complex space. In this note we show that holomorphic mappings φ : B → B belonging to the Schur-Agler class (defined below) satisfy a strengthened form of the Julia-Caratheodory theorem (Theorem 1.9). While the Schur-Agler class has received much attention in the past several years from operator theorists, relatively little seems to be known about the function-theoretic behavior of this class. For many operator theoretic applications, the Schur-Agler classes S(n, 1) and S(n, n) are more appropriate analogues of the unit ball of H∞(D) than are the larger classes Hol(B,D) and Hol(B,B). For example, the Schur-Agler class is a natural setting for multivariable versions of von Neumann’s inequality [5], the Sz.-Nagy dilation theorem [3], commutant lifting theorems [4] and the Nevanlinna-Pick interpolation theorem [1]. Additionally, every self-map of the ball belonging to the Schur-Agler class induces a bounded composition operator on the standard holomorphic function spaces [6], which is not true of general self-maps of the ball. This last fact suggests that mappings in the Schur-Agler class should also enjoy function-theoretic privileges over generic maps of the ball, and is the motivation for this paper. Indeed there seems to be little known about the function theory of S(n,m) apart from what is true generically. Recently Anderson, Dritschel and Rovnyak [2] have established a family of inequalities for derivatives of Schur-Agler functions, though it is not known if these inequalities hold generically. In this paper we show that the Schur-Agler class satisfies a form of the Julia-Caratheodory theorem that is strictly Date: July 8, 2007. Partially supported by NSF grant DMS-0701268. 1