Analytic Schur multipliers

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