Schatten–Herz Classes of Toeplitz Operators on the Generalized Fock Space

Abstract

Let $F^{2}_\varphi$ be the generalized Fock space defined as

$$\begin{aligned} F^2_\varphi =\left\{ f {\text { holomorphic in }} {{\mathbf {C}}^n}: \Vert f\Vert _{2, \varphi }= \left( \int _{{{{\mathbf {C}}^n}}}\left| f(z) e^{-\varphi (z)}\right| ^{2} dv(z)\right) ^\frac{1}{2} <\infty \right\} , \end{aligned}$$

where $\varphi$ is some fixed weight function satisfying $M_1 \omega _0\le d d^c \varphi \le M_2 \omega _0$. Given $0<p\le \infty$ and $0\le q\le \infty$, as an extension of Schatten class Toeplitz operators we introduce the concept of Schatten–Herz class Toeplitz operators $S_{p, q}$ on $F^{2}_\varphi$. We also characterize those positive Borel measures $\mu$ on ${{{\mathbf {C}}^n}}$ for which the induced Toeplitz operators $T_{\mu }$ belong to $S_{p, q}$.