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Hardy Classes and Symbols of Toeplitz Operators

Commun. Math. Anal.
Volume 21, Number 1 (2018), 9 - 22

Hardy Classes and Symbols of Toeplitz Operators

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Abstract

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