{"title":"A Berezin-type map and a class of weighted composition operators","authors":"N. Das","doi":"10.1515/conop-2017-0003","DOIUrl":null,"url":null,"abstract":"Abstract 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 $${t_a}(s) = {{ - ids + (1 - c)} \\over {(1 + c)s + id}} , if a = c + id ∈ 𝔻 c, d ∈ ℝ. For a ∈ 𝔻, define Va:La2(+)→La2(+) ${V_a}:L_a^2({{\\mathbb{C}}_{\\rm{ + }}}) \\to L_a^2({{\\mathbb{C}}_{\\rm{ + }}})$ by (Vag)(s) = (g∘ta)(s)la(s) where la(s)=1-|a|2((1+c)s+id)2 $la(s) = {{1 - {{\\left| a \\right|}^2}} \\over {{{((1 + c)s + id)}^2}}}$ .We look at the action of the class of unitary operators Va, a ∈ 𝔻 on the linear operator L. We establish that Lˆ = L where L⌢=∫𝔻VaLVadA(a) $\\mathord{\\buildrel{\\lower3pt\\hbox{$\\scriptscriptstyle\\frown$}}\\over L} = \\int\\limits_{\\mathbb{D}} {{V_a}L{V_a}dA(a)}$ and dA is the area measure on 𝔻. In fact the map L satisfies the averaging condition L˜(w1)=∫DL˜(ta¯(w1))dA(a),for all w1∈C+ $$\\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{ + }}}}$$ where L˜(w1)=〈Lbw¯1,bw¯1〉 $\\tilde L({w_1}) = \\left\\langle {L{b_{{{\\bar w}_1}}},{b_{{{\\bar w}_1}}}} \\right\\rangle$.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"4 1","pages":"18 - 31"},"PeriodicalIF":0.3000,"publicationDate":"2017-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2017-0003","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Concrete Operators","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1515/conop-2017-0003","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Abstract In this paper we consider the map L defined on the Bergman space La2(+) $L_a^2({{\rm\mathbb{C}}_{\rm{ + }}})$ of the right half plane ℂ+ by (Lf)(w)=πM′(w)∫+(fM′)(s)|bw(s)|2dA˜(s) $(Lf)(w) = \pi M'(w)\int\limits_{{{\rm\mathbb{C}}_{\rm{ + }}}} {\left( {{f \over {M'}}} \right)} (s){\left| {{b_w}(s)} \right|^2}d\tilde A(s)$ where bw¯(s)=1π1+w1+w2Rew(s+w)2 ${b_{\bar w}}(s) = {1 \over {\sqrt \pi }}{{1 + w} \over {1 + w}}{{2{\mathop{Re}\nolimits} w} \over {{{(s + w)}^2}}}$ , s ∈ ℂ+ and Ms=1-s1+s $Ms = {{1 - s} \over {1 + s}}$ . We show that L commutes with the weighted composition operators Wa, a ∈ 𝔻 defined on La2(+) $L_a^2({{\rm\mathbb{C}}_{\rm{ + }}})$ , as Waf=(f∘ta)M′M′∘ta ${W_a}f = (f \circ {t_a}){{M'} \over {M' \circ {t_a}}}$ , f∈La2(+) $f \in L_a^2(\mathbb{C_ + })$ . Here ta(s)=-ids+(1-c)(1+c)s+id $${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 L˜(w1)=∫DL˜(ta¯(w1))dA(a),for all w1∈C+ $$\tilde L({w_1}) = \int\limits_{\mathbb{D}} {\tilde L({t_{\bar a}}({w_1}))dA(a),{\rm{for all }}{w_1} \in {{\rm{C}}_{\rm{ + }}}}$$ where L˜(w1)=〈Lbw¯1,bw¯1〉 $\tilde L({w_1}) = \left\langle {L{b_{{{\bar w}_1}}},{b_{{{\bar w}_1}}}} \right\rangle$.