一个berezin型映射和一类加权复合算子

IF 0.3 Q4 MATHEMATICS
N. Das
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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{ + }}}}$$ where L˜(w1)=〈Lbw¯1,bw¯1〉 $\\tilde L({w_1}) = \\left\\langle {L{b_{{{\\bar w}_1}}},{b_{{{\\bar w}_1}}}} \\right\\rangle$.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"4 1","pages":"18 - 31"},"PeriodicalIF":0.3000,"publicationDate":"2017-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2017-0003","citationCount":"0","resultStr":"{\"title\":\"A Berezin-type map and a class of weighted composition operators\",\"authors\":\"N. Das\",\"doi\":\"10.1515/conop-2017-0003\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract In this paper we consider the map L defined on the Bergman space La2(𝔺+) $L_a^2({{\\\\rm\\\\mathbb{C}}_{\\\\rm{ + }}})$ of the right half plane ℂ+ by (Lf)(w)=πM′(w)∫𝔺+(fM′)(s)|bw(s)|2dA˜(s) $(Lf)(w) = \\\\pi M'(w)\\\\int\\\\limits_{{{\\\\rm\\\\mathbb{C}}_{\\\\rm{ + }}}} {\\\\left( {{f \\\\over {M'}}} \\\\right)} (s){\\\\left| {{b_w}(s)} \\\\right|^2}d\\\\tilde A(s)$ where bw¯(s)=1π1+w1+w2Rew(s+w)2 ${b_{\\\\bar w}}(s) = {1 \\\\over {\\\\sqrt \\\\pi }}{{1 + w} \\\\over {1 + w}}{{2{\\\\mathop{Re}\\\\nolimits} w} \\\\over {{{(s + w)}^2}}}$ , s ∈ ℂ+ and Ms=1-s1+s $Ms = {{1 - s} \\\\over {1 + s}}$ . We show that L commutes with the weighted composition operators Wa, a ∈ 𝔻 defined on La2(𝔺+) $L_a^2({{\\\\rm\\\\mathbb{C}}_{\\\\rm{ + }}})$ , as Waf=(f∘ta)M′M′∘ta ${W_a}f = (f \\\\circ {t_a}){{M'} \\\\over {M' \\\\circ {t_a}}}$ , f∈La2(𝔺+) $f \\\\in L_a^2(\\\\mathbb{C_ + })$ . Here ta(s)=-ids+(1-c)(1+c)s+id $${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}","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

摘要

摘要在本文中,我们考虑到地图上定义L伯格曼空间水(𝔺+)美元L_a ^ 2 ({{\ rm \ mathbb {C}} _ {\ rm { + }}})$ 正确的半平面ℂ由(低频)+ (w) =πM ' (w)∫𝔺+ (fM) (s) | bw (s) | 2 da˜(s)(低频)美元(w) = \πM ' int (w) \ \ limits_ {{{rm \ \ mathbb {C}} _ {\ rm { + }}}} {\ 左({{f \ / {M '}}} \右)}(s){\左| {{b_w} (s)} \右| ^ 2}d \波浪号(s)美元在bw¯(s) = 1π1 + w1 + w2Rew (s + w) 2 $ {b_{\酒吧w}} (s) ={1 \ /{\√6 \π}}{{1 + w} \ / {1 + w}} {{2 {\ mathop{你}\长成具}w} \ / {{{(s + w)} ^ 2}}} $,年代∈ℂ+女士和女士= 1 s1 + s $ ={{1,}在{1 + s}} \ $。我们表明,L通勤加权复合算子与佤邦,一个∈𝔻定义在水(𝔺+)美元L_a ^ 2 ({{\ rm \ mathbb {C}} _ {\ rm { + }}})$ , Waf = (f∘ta) M是“∘ta $ {W_a} f = (f \保监会{t_a}) {{} \ / {M M \保监会{t_a}}} $ f∈水(𝔺+)$ f \ L_a ^ 2 (\ mathbb {C_ +})美元。这里的助教(s) = id + (1 - c) (1 + c) s + id $ $ {t_a} (s) = {{- id + (1 - c)} \ / {(1 + c) s + id}},如果= c + id∈𝔻c, d∈ℝ。弗吉尼亚州的∈𝔻,定义:水(𝔺+)→水(𝔺+)$ {V_a}: L_a ^ 2 ({{\ mathbb {C}} _ {\ rm { + }}}) \ 对L_a ^ 2 ({{\ mathbb {C}} _ {\ rm { + }}})$ (流浪者)(s) = (g∘ta) (s)拉(s)在洛杉矶(s) = 1 - | | 2 ((1 + c) s + id) 2美元拉(s) ={{1 -{{\左| \右|}^ 2}}\ / {{{((1 + c) s + id)} ^ 2}}} $。我们看看单一运营商Va的动作类的,我们建立了L∈L = L,其中L =∫𝔻VaLVadA(a) $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over L} = \int\limits_{\mathbb{D}} {{V_a}L{V_a}dA(a)}$, dA是在上的面积测度。事实上地图L满足平均条件下L˜(w1) =∫DL˜(ta¯(w1)) dA (a),对所有w1∈C + L $ $ \波浪号({w_1}) = \ int \ limits_ {\ mathbb {D}}{\波浪号L ({t_{\酒吧}}识别({w_1})) dA (a), {\ rm所有}{}{w_1} \在{{\ rm {C}} _ {\ rm { + }}}}$$ 在L˜(w1) = <激光焊¯1,bw¯1 > \波浪号L ({w_1}) =美元\左\ langle {L {b_{{{酒吧\ w} _1}}}, {b_{{{酒吧\ w} _1}}}} \ \纠正美元。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
A Berezin-type map and a class of weighted composition operators
Abstract In this paper we consider the map L defined on the Bergman space La2(𝔺+) $L_a^2({{\rm\mathbb{C}}_{\rm{ + }}})$ of the right half plane ℂ+ by (Lf)(w)=πM′(w)∫𝔺+(fM′)(s)|bw(s)|2dA˜(s) $(Lf)(w) = \pi M'(w)\int\limits_{{{\rm\mathbb{C}}_{\rm{ + }}}} {\left( {{f \over {M'}}} \right)} (s){\left| {{b_w}(s)} \right|^2}d\tilde A(s)$ where bw¯(s)=1π1+w1+w2Rew(s+w)2 ${b_{\bar w}}(s) = {1 \over {\sqrt \pi }}{{1 + w} \over {1 + w}}{{2{\mathop{Re}\nolimits} w} \over {{{(s + w)}^2}}}$ , s ∈ ℂ+ and Ms=1-s1+s $Ms = {{1 - s} \over {1 + s}}$ . We show that L commutes with the weighted composition operators Wa, a ∈ 𝔻 defined on La2(𝔺+) $L_a^2({{\rm\mathbb{C}}_{\rm{ + }}})$ , as Waf=(f∘ta)M′M′∘ta ${W_a}f = (f \circ {t_a}){{M'} \over {M' \circ {t_a}}}$ , f∈La2(𝔺+) $f \in L_a^2(\mathbb{C_ + })$ . Here ta(s)=-ids+(1-c)(1+c)s+id $${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〉 $\tilde L({w_1}) = \left\langle {L{b_{{{\bar w}_1}}},{b_{{{\bar w}_1}}}} \right\rangle$.
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来源期刊
Concrete Operators
Concrete Operators MATHEMATICS-
CiteScore
1.00
自引率
16.70%
发文量
10
审稿时长
22 weeks
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