Hardy Classes and Symbols of Toeplitz Operators

The purpose of this paper is to study functions in the unit disk $\mathbb D$ through the family of Toeplitz operators $\{T_{φdσ_{t}}\}_{t∈[0,1)}$, where $T_{φdσ_{t}}$ is the Toeplitz operator acting the Bergman space of $\mathbb D$ and where $dσ_t$ is the Lebesgue measure in the circle $tS^1$. In particular for $1\le p \lt \infty$ we characterize the harmonic functions $φ$ in the Hardy space $h^{p}(\mathbb D)$ by the growth in $t$ of the $p$-Schatten norms of $T_{φdσ_{t}}$. We also study the dependence in $t$ of the norm operator of $T_{adσ_{t}}$ when $a∈H^p_{at}$, the atomic Hardy space in the unit circle with $1/2 \lt p \le 1$.