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

In this paper we consider the map L defined on the Bergman space La2(𝔺+)$L_a^2({{\rm\mathbb{C}}_{\rm{ + }}})$ of the right half plane ℂ+ by (Lf)(w)=πM′(w)∫𝔺+(fM′)(s)|bw(s)|2dA˜(s)$(Lf)(w) = \pi M'(w)\int\limits_{{{\rm\mathbb{C}}_{\rm{ + }}}} {\left( {{f \over {M'}}} \right)} (s){\left| {{b_w}(s)} \right|^2}d\tilde A(s)$ where bw¯(s)=1π1+w1+w2Rew(s+w)2${b_{\bar w}}(s) = {1 \over {\sqrt \pi }}{{1 + w} \over {1 + w}}{{2{\mathop{Re}\nolimits} w} \over {{{(s + w)}^2}}}$, s ∈ ℂ+ and Ms=1-s1+s$Ms = {{1 - s} \over {1 + s}}$. We show that L commutes with the weighted composition operators Wa, a ∈ 𝔻 defined on La2(𝔺+)$L_a^2({{\rm\mathbb{C}}_{\rm{ + }}})$, as Waf=(f∘ta)M′M′∘ta${W_a}f = (f \circ {t_a}){{M'} \over {M' \circ {t_a}}}$, f∈La2(𝔺+)$f \in L_a^2(\mathbb{C_ + })$. Here ta(s)=-ids+(1-c)(1+c)s+id$${t_a}(s) = {{ - ids + (1 - c)} \over {(1 + c)s + id}}, if a = c + id ∈ 𝔻 c, d ∈ ℝ. For a ∈ 𝔻, define Va:La2(𝔺+)→La2(𝔺+)${V_a}:L_a^2({{\mathbb{C}}_{\rm{ + }}}) \to L_a^2({{\mathbb{C}}_{\rm{ + }}})$by (Vag)(s) = (g∘ta)(s)la(s) where la(s)=1-|a|2((1+c)s+id)2$la(s) = {{1 - {{\left| a \right|}^2}} \over {{{((1 + c)s + id)}^2}}}$.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 L⌢=∫𝔻VaLVadA(a)$\mathord{\buildrel{\lower3pt\hbox{$\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

Abstract licence: CC BY

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