Beurling-Malliavin theory for Toeplitz kernels

Abstract

We consider the family of Toeplitz operators $T_{J{\bar{S}}^a}$ acting in the Hardy space $H^2$ in the upper halfplane; $J$ and $S$ are given meromorphic inner functions, and $a$ is a real parameter. In the case where the argument of $S$ has a power law type behavior on the real line, we compute the critical value $$c(J,S)=\inf \left\{a:\ker T_{J{\bar{S}}^a}\ne 0\right\}.$$ The formula for $c(J,S)$ generalizes the Beurling-Malliavin theorem on the radius of completeness for a system of exponentials.

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