Two-parameter Toeplitz-type operators between Fock spaces

We introduce and study a two-parameter family of Toeplitz-type operators $T_{\phi }^{(s,t)}$ on the Fock space $F^p(C)$ for $0<p\le \mathrm{\infty }$. For all $0<p,q\le \mathrm{\infty }$ we obtain sharp conditions for $\phi$ such that $T_{\phi }^{(s,t)}:F^p(C)\to F^q(C)$ is bounded or compact. In the Hilbert space case $T_{\phi }^{(s,t)}:F^2(C)\to F^2(C)$, we also study the Berezin transform of $T_{\phi }^{(s,t)}$ and membership of $T_{\phi }^{(s,t)}$ in Schatten classes. Furthermore, we find a class of noncompact operators whose Berezin transforms nevertheless vanish at infinity.