## Functiones et Approximatio Commentarii Mathematici

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

Artur Michalak

#### 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 $\mathbb{D}$.

#### Note

Dedicated to Włodzimierz Staś on the occasion of his 75th birthday

#### Article information

Source
Funct. Approx. Comment. Math., Volume 28 (2000), 211-220.

Dates
First available in Project Euclid: 29 September 2018

https://projecteuclid.org/euclid.facm/1538186697

Digital Object Identifier
doi:10.7169/facm/1538186697

Mathematical Reviews number (MathSciNet)
MR1824006

Zentralblatt MATH identifier
0987.47020

Keywords
Toeplitz operators Bergman spaces

#### Citation

Michalak, Artur. Essential norms of Toeplitz operators on Bergman-Hardy spaces on the unit disk. Funct. Approx. Comment. Math. 28 (2000), 211--220. doi:10.7169/facm/1538186697. https://projecteuclid.org/euclid.facm/1538186697