Abstract:
In this paper, enlightened by Isralowitz’s works on the properties of weakly localized operators on Fock spaces, we use Berezin transform as a key tool to investigate boundedness and compactness of Toeplitz operators
Tφ with symbols
φ in BMO on the Fock spaces F_α^p(1<
p<+∞). We extend properties of
Tφ on F_α^2 to the more general F_α^p setting, yielding two main results:
Tφ is a bounded operator on F_α^p if and only if the Berezin transform \widetilde\varphi of
φ is bounded,
Tφ is a compact operator on F_α^p if and only if \widetilde\varphi∈C
0(\mathbbC). If
φ is harmonic function of BMO, then
Tφ is bounded on F_α^p if and only if
φ is bounded,
Tφ is compact on if and only if
φ∈C
0(\mathbbC).