$dist$-formulas and Toeplitz operators

Abstract

The distance from the nonconstant function $\varphi$ in $L^{\infty}% (\mathbb{T})$ to the set $\mathcal{F}_{\text{const}}$ of all constant functions is estimated in terms of Hankel operators on the Hardy space $H^{2}(\mathbb{D})$ over the unit disk $\mathbb{D}=\left\{ z\in \mathbb{C}:\left\vert z\right\vert \in [0, 1)\right\} $. We give a sufficient condition ensuring the equality $dist(\varphi,\mathcal{F}_{\text{const}})=\left\Vert \varphi\right\Vert_{L^{\infty}}$. Some other $dist$-formulas are also discussed.