Schatten-Herz Operators, Berezin Transform and Mixed Norm Spaces

Abstract

For p, q > 0 we study operators T on the Bergman space ${A_{2}(\mathbb{D)}}$ in the disk such that ${\left(\sum_{j}\Vert T\Delta_{j}\Vert_{p}^{q}\right)^{1/q}<\infty,}$ where the norms ${\Vert\cdot\Vert_{p}}$ are in the Schatten class S _p (A ₂), the projection ${\Delta_{j}f=\sum_{n\in I_{j}}a_{n}z^{n}}$ for ${f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}}$ and ${I_{j}=[2^{j}-1,2^{j+1} )\cap(\mathbb{N}\cup\{0\})}$ for ${j\in\mathbb{N}\cup\{0\}.}$ We consider the relation of this property with mixed norms of the Berezin transform of T and of the related function ${f_{T}(z)={\Vert}T(k_{z})\Vert}$ where k _z is the normalized Bergman kernel. These classes of operators denoted by S(p, q) are closely related when assumed to be positive with other sets of operators, like the class of positive operators on A ₂ for which ${\left(\sum_{j\geq0}(\sum_{n\in I_{j}}|\left\langle T^pe_{n},e_{n}\right\rangle |)^{q/p}\right)^{1/q}<\infty}$ , where ${\{e_{n}\}_{n\geq0}}$ is the canonical basis of A ₂; also we study the relation of Toeplitz operators in S(p, q) with the Schatten-Herz classes, where the decomposition is through dyadic annuli of the domain ${\mathbb{D}}$ .