Essential norms of Toeplitz operators on Bergman-Hardy spaces on the unit disk

Abstract

For a Borel measure $\mu$ on $[0, 1]$ with $1 \in \mathrm{supp}(\mu)$ we consider the Toeplitz operators $T_f$ with $f \in L^{\infty}(\mu \bigotimes \lambda)$ on the space $H^{2}(\mu)$ consisting of all holomorphic on $\mathbb{D}$ functions in the Lebegue class $L^{2}(\mu \bigotimes \lambda)$ on $\bar{\mathbb{D}}$. We show that for every Bergman-Hardy space $H^{2}(\mu)$ the following estimations hold: $\mathrm{dist}(T_{f}, \mathcal{K}(H^{2}(\mu))) \geqslant \vert f(t_0) \vert$ if $f$ is continuous at point $t_0 \in \mathbb{T}$, and $\mathrm{dist}(T_{f}, \mathcal{K}(H^{2}(\mu))) = \mathrm{sup}\{ \vert f(z) \vert : z \in \mathbb{D} \}$ if $f$ is a bounded harmonic function on $\mathbb{D}$.