DYNAMICS OF PROPERTIES OF TOEPLITZ OPERATORS ON THE UPPER HALF-PLANE: PARABOLIC CASE

We consider Toeplitz operators ${\mathrm{T}}_{\mathrm{a}}^{\left(\mathrm{\lambda }\right)}$ acting on the weighted Bergman spaces ${\mathrm{A}}_{\mathrm{\lambda }}^{2}\left(\mathrm{\Pi }\right)$, λ ∈ [0, ∞), over the upper half-plane Π, whose symbols depend on y = Im z. Motivated by the Berezin quantization procedure we study the dependence of the properties of such operators on the weight λ and, in particular, under the limit procedure λ → ∞.