On Reachable Families of the Loewner Differential Equation in Several Complex Variables

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• 作者：Mihai Iancu
• 关键词：Bergman space ; Bochner space ; Carathéodory family ; Loewner differential equation ; Reachable family ; Weak topology ; Primary 32H ; 30C45 ; Secondary 28B05 ; 32A25 ; 93B03
• 刊名：Complex Analysis and Operator Theory
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：10
• 期：2
• 页码：353-368
• 全文大小：506 KB
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• 作者单位：Mihai Iancu (1)
  
  1. Faculty of Mathematics and Computer Science, Babeş-Bolyai University, 1 M. Kogălniceanu Str., 400084, Cluj-Napoca, Romania
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Operator Theory
  Analysis
  
• 出版者：Birkh盲user Basel
• ISSN：1661-8262

文摘

Graham, Hamada, Kohr and Kohr studied the normalized time \(T\) reachable families \(\widetilde{\mathcal {R}}_T(id_{{\mathbb {B}}^n},\Omega )\) of the Loewner differential equation, which are generated by the Carathéodory mappings with values in a subfamily \(\Omega \) of the Carathéodory family \({\mathcal {N}}_A\) for the Euclidean unit ball \({\mathbb {B}}^n\), where \(A\) is a linear operator with \(k_+(A)<2m(A)\) (\(k_+(A)\) is the Lyapunov index of \(A\) and \(m(A)=\min \{\mathfrak {R}\left\langle Az,z\right\rangle \big |z\in {\mathbb {C}}^n,\Vert z\Vert =1\}\)). They obtained some compactness and density results, as generalizations of related results due to Roth, and conjectured that if \(\Omega \) is compact and convex, then \(\widetilde{\mathcal {R}}_T(id_{{\mathbb {B}}^n},\Omega )\) is compact and \(\widetilde{\mathcal {R}}_T(id_{{\mathbb {B}}^n},ex\,\Omega )\) is dense in \(\widetilde{\mathcal {R}}_T(id_{{\mathbb {B}}^n},\Omega )\), where \(ex\,\Omega \) denotes the corresponding set of extreme points and \(T\in [0,\infty ]\). We confirm this, by embedding the Carathéodory mappings in a suitable Bochner space. Keywords Bergman space Bochner space Carathéodory family Loewner differential equation Reachable family Weak topology