Schatten class Toeplitz operators on generalized Fock spaces (Q401363)

Let \(\phi \in C^2(\mathbb{C}^n)\) be a real valued function on \(\mathbb{C}^n\) that satisfies, for positive constants \(c\) and \(C\), NEWLINE\[NEWLINEcw_0<d d^c \phi<Cw_0,NEWLINE\]NEWLINE where \(d\) is the usual exterior derivative, \(d^c= \frac{i}{4}(\overline{\partial}-\partial)\), and \(w_0=dd^c |\cdot|^2\) is the standard Euclidean Kähler form. The generalized Fock space \(F^{2}_{\phi}\) consists of all entire functions \(f\) such that NEWLINE\[NEWLINE\int_{\mathbb{C}^n} |f(z)|^2 e^{-2 \phi(z)} dv(z)<\infty.NEWLINE\]NEWLINE Let \(P\) denote the orthogonal projection of \(L^2(e^{-2\phi} dv)\) onto \(F^{2}_{\phi}\). For a positive measure \(\mu\), the Toeplitz operator \(T_{\mu}\) on \(F^{2}_{\phi}\) is defined as NEWLINE\[NEWLINET_{\mu}f(z)=\int_{\mathbb{C}^n} K(z, w) f(w) e^{-2\phi(w)} d\mu(w),NEWLINE\]NEWLINE where \(K\) is the reproducing kernel of \(F^{2}_{\phi}\). In addition, the authors assume that \(\mu\) satisfies, for all \(\gamma>0\) and \(z\in\mathbb{C}^n\), the condition NEWLINE\[NEWLINE\int_{\mathbb{C}^n} e^{-\gamma |z-w|} d\mu(w)<\infty.NEWLINE\]NEWLINE The authors characterize the Schatten class membership of the Toeplitz operators \(T_{\mu}\) in terms of the heat (Berezin) transform \(\tilde{\mu}\), and in terms of the averaging function \(\mu(B(\cdot, r))\). The proofs of their characterizations are divided into the cases \(0<p\leq 1\) and \(1\leq p<\infty\), the proofs rely on some special estimates on the behavior of the reproducing kernel.