The generalized Toeplitz operators on the Fock space $F_{\alpha }^{2}$

Xu, Chunxu ; Yu, Tao

Czechoslovak Mathematical Journal, Tome 74 (2024) no. 1, pp. 231-246.

Résumé

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 $.

Détail

DOI : 10.21136/CMJ.2024.0066-23

Classification : 30H20, 47B35
Keywords: generalized Toeplitz operator; boundedness; compactness; Schatten class; Fock space