The generalized Toeplitz operators on the Fock space \(F_\alpha ^2\)

Let μ be a positive Borel measure on the complex plane ℂⁿ and let j = (j₁, …, j_n) with j_i ∈ ℕ. We study the generalized Toeplitz operators \(T_\mu ^{(j)}\) $$T_{\mu }^{(j)}$$ on the Fock space \(F_\alpha ^2\) $$F_{\alpha }^2$$ . We prove that \(T_\mu ^{(j)}\) $$T_{\mu }^{(j)}$$ is bounded (or compact) on \(F_\alpha ^2\) $$F_{\alpha }^2$$ if and only if μ is a Fock-Carleson measure (or vanishing Fock-Carleson measure). Furthermore, we give a necessary and sufficient condition for \(T_\mu ^{(j)}\) $$T_{\mu }^{(j)}$$ to be in the Schatten p-class for 1 ⩽ p < ∞.