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Keywords:
generalized Toeplitz operator; boundedness; compactness; Schatten class; Fock space
generalized Toeplitz operator; boundedness; compactness; Schatten class; Fock space
Summary:
Let $\mu $ be a positive Borel measure on the complex plane $\mathbb {C}^n$ and let $j=(j_1,\cdots ,j_n)$ with $j_i\in \mathbb {N}$. We study the generalized Toeplitz operators $T_{\mu }^{(j)}$ on the Fock space $F_{\alpha }^{2}$. We prove that $T_{\mu }^{(j)}$ is bounded (or compact) on $F_{\alpha }^{2}$ if and only if $\mu $ is a Fock-Carleson measure (or vanishing Fock-Carleson measure). Furthermore, we give a necessary and sufficient condition for $T_{\mu }^{(j)}$ to be in the Schatten $p$-class for $1\leq p<\infty $.
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