Geodesic
The generalized Toeplitz operators on the Fock space $F_{\alpha }^{2}$
Czechoslovak Mathematical Journal, Tome 74 (2024) no. 1, pp. 231-246
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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 $.
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 $.
DOI : 10.21136/CMJ.2024.0066-23
Classification : 30H20, 47B35
Keywords: generalized Toeplitz operator; boundedness; compactness; Schatten class; Fock space 
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Xu, Chunxu; Yu, Tao. The generalized Toeplitz operators on the Fock space $F_{\alpha }^{2}$. Czechoslovak Mathematical Journal, Tome 74 (2024) no. 1, pp. 231-246. doi: 10.21136/CMJ.2024.0066-23

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