Analytic Schur multipliers

We study in this paper analytic Schur multipliers on C₊² and D², i.e. Schur multipliers on R² and T² that are boundary-value functions of functions analytic in C₊² and D². Such Schur multipliers are important when studying properties of functions of maximal dissipative operators and contractions under perturbation. We show that if the boundary-value function of a Schur multiplier has certain regularity properties, then it can be represented as an element of the Haagerup tensor product of spaces of analytic functions with similar regularity properties.