On Toeplitz operators between Fock spaces (Q469581)

Let \(d\nu\) be the usual Lebesgue measure on \(\mathbb{C}^{n}\). For \(\alpha\) a positive parameter and \(1\leq p\leq \infty\), the Fock spaces \(\mathcal{F}_{\alpha}^{p}\) consist of entire functions \(f\) for which \[ \|f\|_{p, \alpha}^{p}=\left (\frac{\alpha p}{2\pi}\right )^{n}\int_{\mathbb{C}^{n}}|f(z)|^{p}e^{-\frac{\alpha p}{2}|z|^{2}}d\nu(z) < \infty, \text{ for } 1\leq p<\infty, \] and \[ \|f\|_{\infty, \alpha}=\sup_{z\in \mathbb{C}^{n}}|f(z)|e^{-\frac{\alpha p}{2}|z|^{2}}< \infty. \] Let \(\mu\) be a non-negative Borel measure on \(\mathbb{C}^{n}\). Then the author characterizes the boundedness and compactness of the Toeplitz operator \(T_{\mu}:\mathcal{F}_{\alpha}^{p}\to \mathcal{F}_{\alpha}^{\infty}\) and \(T_{\mu}:\mathcal{F}_{\alpha}^{\infty}\to \mathcal{F}_{\alpha}^{p}\) for \(1\leq p<\infty\). Furthermore, for \(1\leq q\leq \infty\), the author characterizes the boundedness and compactness for the operators \(T_{\mu}:\mathcal{F}_{\alpha}^{1}\to \mathcal{F}_{\alpha}^{q}\) and \(T_{\mu}:\mathcal{F}_{\alpha}^{q}\to \mathcal{F}_{\alpha}^{1}\). The author also obtains an asymptotic estimate for the norms of these operators.