The Berezin Transformation on L 2 (U+)

Let L 2 a (U+) be the Bergman space of the upper half plane U+. In this paper, we consider the integral operator H from L 2 (U+) into L 2 (U+) defined by (Hf)(w) = fe(w) = Z U+ f(s)|dw(s)| 2 dAe(s), w ∈ U+, where dw(s) = 1 √ π w + i w − i (−2i)Im w (s + w) 2 and dAe is the area measure on U+. We refer the map H as the Berezin transformation defined on L 2 (U+). We have derived various algebraic properties of the operator and showed that ||H|| ≤ 3π 4 considered as an operator on L 2 a (U+).