Toeplitz Operators in the Herglotz Space

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• 作者：Grigori Rozenblum ; Nikolai Vasilevski
• 关键词：Bergman type spaces ; Helmholtz equation ; Toeplitz operators
• 刊名：Integral Equations and Operator Theory
• 出版年：2016
• 出版时间：November 2016
• 年：2016
• 卷：86
• 期：3
• 页码：409-438
• 全文大小：711 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  
• 出版者：Birkh盲user Basel
• ISSN：1420-8989
• 卷排序：86

文摘

We define and study Toeplitz operators in the space of Herglotz solutions of the Helmholtz equation in \({\mathbb R}^d\). Since the most traditional definition of Toeplitz operators via Bergman-type projection is not available here, we use the approach based upon the reproducing kernel nature of the Herglotz space and sesquilinear forms, which results in a meaningful theory. For two important patterns of sesquilinear forms we discuss a number of properties, including the uniqueness of determining the symbols from the operator, the finite rank property, the conditions for boundedness and compactness, spectral properties, certain algebraic relations.