Toeplitz Operators on Fock Spaces $${F^{p}(\varphi)}$$ F p ( φ )

Given \({\varphi\in \verb"C"^2(\textbf{C}^n)}\) satisfying \({dd^{c}\varphi\simeq \omega_0}\), 0 < p < ∞, let \({F^p(\varphi)}\) be the generalized Fock space of all holomorphic functions f on \({{\mathbf C}^n}\) for which the Fock norm $$\|f\|_{p, \varphi}=\left(\,\int_{{\mathbf C}^n} \left|f(z)\right|^{p}e^ {-p\varphi(z)}dv(z)\right)^{\frac{1}{p}} < \infty. $$ While \({\varphi(z)=\frac{1}{2}|z|^2}\), \({F^{p}(\varphi)}\) is the classical Fock space Fp. In this paper, for all possible 0 < p,q < ∞ we characterize those positive Borel measures μ on \({{\mathbf C}^n}\) for which the induced Toeplitz operators Tμ are bounded (or compact) from one generalized Fock spaces \({F^p(\varphi)}\) to another \({F^q(\varphi)}\). With symbols \({g\in BMO}\), we obtain Zorborska’s criterion for boundedness (or compactness) of Toeplitz operators Tg on Fp, our work extends the known results on F2. Toeplitz operators on p-th Fock space with 0 < p < 1 have not been studied before, even in the simplest case that \({\varphi(z)=\frac{1}{2}|z|^2}\). Our analysis shows a significant difference between Bergman spaces and Fock spaces.