{"title":"Some questions about complex harmonic functions","authors":"Luis E. Benítez-Babilonia, Raúl Felipe","doi":"10.1007/s00605-024-01956-0","DOIUrl":null,"url":null,"abstract":"<p>In this paper, we propose composition products in the class of complex harmonic functions so that the composition of two such functions is again a complex harmonic function. From here, we begin the study of the iterations of the functions of this class showing briefly its potential to be a topic of future research. In parallel, we define and study composition operators on a Hardy type space denoted by <span>\\(HH^{2}(\\mathbb {D})\\)</span> of complex harmonic functions also introduced for us in the present work. The symbols of these composition operators have of form <span>\\(\\chi +\\overline{\\pi }\\)</span> where <span>\\(\\chi ,\\pi \\)</span> are analytic functions from <span>\\(\\mathbb {D}\\)</span> into <span>\\(\\mathbb {D}\\)</span>. We also analyze the space of bounded linear operators on <span>\\(HH^{2}(\\mathbb {D})\\)</span>.</p>","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Monatshefte für Mathematik","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s00605-024-01956-0","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we propose composition products in the class of complex harmonic functions so that the composition of two such functions is again a complex harmonic function. From here, we begin the study of the iterations of the functions of this class showing briefly its potential to be a topic of future research. In parallel, we define and study composition operators on a Hardy type space denoted by \(HH^{2}(\mathbb {D})\) of complex harmonic functions also introduced for us in the present work. The symbols of these composition operators have of form \(\chi +\overline{\pi }\) where \(\chi ,\pi \) are analytic functions from \(\mathbb {D}\) into \(\mathbb {D}\). We also analyze the space of bounded linear operators on \(HH^{2}(\mathbb {D})\).