Analytic Schur multipliers

We study in this paper analytic Schur multipliers on C + 2 C+^2 and D 2 D^2, i. e. Schur multipliers on R 2 R^2 and T 2 T^2 that are boundary-value functions of functions analytic in C + 2 C+^2 and D 2 D^2. 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.