Abstract
In this paper, we study Toeplitz operators T μ from one Fock space \({F^{p}_{\alpha}}\) to another \({F^{q}_{\alpha}}\) for 1 < p, q < ∞ with positive Borel measures μ as symbols. We characterize the boundedness (and compactness) of \({T_\mu: F^{p}_{\alpha} \to F^{q}_{\alpha}}\) in terms of the averaging function \({\widehat{\mu}_r}\) and the t-Berezin transform \({\widetilde{\mu}_t}\) respectively. Quite differently from the Bergman space case, we show that T μ is bounded (or compact) from \({F^{p}_{\alpha}}\) to \({F^{q}_{\alpha}}\) for some p ≤ q if and only if T μ is bounded (or compact) from \({F^{p}_{\alpha}}\) to \({F^{q}_{\alpha}}\) for all p ≤ q. In order to prove our main results on T μ , we introduce and characterize (vanishing) (p, q)-Fock Carleson measures on C n.
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This work was completed with the support of National Natural Science Foundation of China (10771064), Natural Science Foundation of Zhejiang province (Y7080197, Y6090036, Y6100219) and Foundation of Creative Group in Colleges and Universities of Zhejiang Province (T200924).
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Hu, Z., Lv, X. Toeplitz Operators from One Fock Space to Another. Integr. Equ. Oper. Theory 70, 541–559 (2011). https://doi.org/10.1007/s00020-011-1887-y
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DOI: https://doi.org/10.1007/s00020-011-1887-y