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 \(S_{p, q}\).