On the Dilation of Truncated Toeplitz Operators

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• 作者：Eungil Ko ; Ji Eun Lee
• 关键词：Dilation of truncated Toeplitz operator ; Normal operator ; Hyponormal operator ; Selfadjoint operator
• 刊名：Complex Analysis and Operator Theory
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：10
• 期：4
• 页码：815-833
• 全文大小：527 KB
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• 作者单位：Eungil Ko (1)
  Ji Eun Lee (2)
  
  1. Department of Mathematics, Ewha Womans University, Seoul, 120-750, Korea
  2. Department of Mathematics-Applied Statistics, Sejong University, Seoul, 143-747, Korea
  
• 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

An operator \(S_{\varphi ,\psi }^{u}\in \mathcal {L}(L^2)\) is called the dilation of a truncated Toeplitz operator if for two symbols \(\varphi ,\psi \in L^{\infty }\) and an inner function u, $$\begin{aligned} S_{\varphi ,\psi }^{u}f=\varphi P_uf+\psi Q_uf \end{aligned}$$holds for \(f\in {L}^{2}\) where \(P_{u}\) denotes the orthogonal projection of \(L^2\) onto the model space \(\mathcal { K}_{u}^2=H^2{\ominus }{{u}H^2}\) and \(Q_u=I-P_u.\) In this paper, we study properties of the dilation of truncated Toeplitz operators on \(L^{2}\). In particular, we provide conditions for the dilation of truncated Toeplitz operators to be normal. As some applications, we give several examples of such operators. Keywords Dilation of truncated Toeplitz operator Normal operator Hyponormal operator Selfadjoint operator Mathematics Subject Classification Primary 47B35 47B15 47B20 Communicated by Matthias Langer.