A Berezin-type map and a class of weighted composition operators

In this paper we consider the map L defined on the Bergman space $L_a^2({{\rm\mathbb{C}}_{\rm{ + }}})$ of the right half plane ℂ+ by $(Lf)(w) = \pi M'(w)\int\limits_{{{\rm\mathbb{C}}_{\rm{ + }}}} {\left( {{f \over {M'}}} \right)} (s){\left| {{b_w}(s)} \right|^2}d\tilde A(s)$ where ${b_{\bar w}}(s) = {1 \over {\sqrt \pi }}{{1 + w} \over {1 + w}}{{2{\mathop{Re}\nolimits} w} \over {{{(s + w)}^2}}}$ , s ∈ ℂ+ and $Ms = {{1 - s} \over {1 + s}}$ . We show that L commutes with the weighted composition operators Wa, a ∈ 𝔻 defined on $L_a^2({{\rm\mathbb{C}}_{\rm{ + }}})$ , as ${W_a}f = (f \circ {t_a}){{M'} \over {M' \circ {t_a}}}$ , $f \in L_a^2(\mathbb{C_ + })$ . Here $${t_a}(s) = {{ - ids + (1 - c)} \over {(1 + c)s + id}} , if a = c + id ∈ 𝔻 c, d ∈ ℝ. For a ∈ 𝔻, define ${V_a}:L_a^2({{\mathbb{C}}_{\rm{ + }}}) \to L_a^2({{\mathbb{C}}_{\rm{ + }}})$ by (Vag)(s) = (g∘ta)(s)la(s) where $la(s) = {{1 - {{\left| a \right|}^2}} \over {{{((1 + c)s + id)}^2}}}$ .We look at the action of the class of unitary operators Va, a ∈ 𝔻 on the linear operator L. We establish that Lˆ = L where $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over L} = \int\limits_{\mathbb{D}} {{V_a}L{V_a}dA(a)}$ and dA is the area measure on 𝔻. In fact the map L satisfies the averaging condition $$\tilde L({w_1}) = \int\limits_{\mathbb{D}} {\tilde L({t_{\bar a}}({w_1}))dA(a),{\rm{for all }}{w_1} \in {{\rm{C}}_{\rm{ + }}}}$$ where $\tilde L({w_1}) = \left\langle {L{b_{{{\bar w}_1}}},{b_{{{\bar w}_1}}}} \right\rangle$.