Analytic Schur multipliers

TL;DR

This paper investigates analytic Schur multipliers on bidimensional complex domains, establishing their representation via the Haagerup tensor product under certain boundary regularity conditions, with implications for operator theory.

Contribution

It introduces a representation theorem for boundary-value functions of analytic Schur multipliers as elements of the Haagerup tensor product, under regularity assumptions.

Findings

Representation of Schur multipliers via Haagerup tensor product

Boundary regularity conditions imply specific tensor product structure

Applications to perturbation theory of operators

Abstract

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