Ian Graham, Hidetaka Hamada, Gabriela Kohr, Mirela Kohr
{"title":"Loewner PDE in Infinite Dimensions","authors":"Ian Graham, Hidetaka Hamada, Gabriela Kohr, Mirela Kohr","doi":"10.1007/s40315-024-00536-5","DOIUrl":null,"url":null,"abstract":"<p>In this paper, we prove the existence and uniqueness of the solution <i>f</i>(<i>z</i>, <i>t</i>) of the Loewner PDE with normalization <span>\\(Df(0,t)=e^{tA}\\)</span>, where <span>\\(A\\in L(X,X)\\)</span> is such that <span>\\(k_+(A)<2m(A)\\)</span>, on the unit ball of a separable reflexive complex Banach space <i>X</i>. In particular, we obtain the biholomorphicity of the univalent Schwarz mappings <i>v</i>(<i>z</i>, <i>s</i>, <i>t</i>) with normalization <span>\\(Dv(0,s,t)=e^{-(t-s)A}\\)</span> for <span>\\(t\\ge s\\ge 0\\)</span>, where <span>\\(m(A)>0\\)</span>, which satisfy the semigroup property on the unit ball of a complex Banach space <i>X</i>. We further obtain the biholomorphicity of <i>A</i>-normalized univalent subordination chains under some normality condition on the unit ball of a reflexive complex Banach space <i>X</i>. We prove the existence of the biholomorphic solutions <i>f</i>(<i>z</i>, <i>t</i>) of the Loewner PDE with normalization <span>\\(Df(0,t)=e^{tA}\\)</span> on the unit ball of a separable reflexive complex Banach space <i>X</i>. The results obtained in this paper give some positive answers to the open problems and conjectures proposed by the authors in 2013.</p>","PeriodicalId":49088,"journal":{"name":"Computational Methods and Function Theory","volume":"37 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Computational Methods and Function Theory","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s40315-024-00536-5","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we prove the existence and uniqueness of the solution f(z, t) of the Loewner PDE with normalization \(Df(0,t)=e^{tA}\), where \(A\in L(X,X)\) is such that \(k_+(A)<2m(A)\), on the unit ball of a separable reflexive complex Banach space X. In particular, we obtain the biholomorphicity of the univalent Schwarz mappings v(z, s, t) with normalization \(Dv(0,s,t)=e^{-(t-s)A}\) for \(t\ge s\ge 0\), where \(m(A)>0\), which satisfy the semigroup property on the unit ball of a complex Banach space X. We further obtain the biholomorphicity of A-normalized univalent subordination chains under some normality condition on the unit ball of a reflexive complex Banach space X. We prove the existence of the biholomorphic solutions f(z, t) of the Loewner PDE with normalization \(Df(0,t)=e^{tA}\) on the unit ball of a separable reflexive complex Banach space X. The results obtained in this paper give some positive answers to the open problems and conjectures proposed by the authors in 2013.
期刊介绍:
CMFT is an international mathematics journal which publishes carefully selected original research papers in complex analysis (in a broad sense), and on applications or computational methods related to complex analysis. Survey articles of high standard and current interest can be considered for publication as well.