Symmetric Pairs and Self-Adjoint Extensions of Operators, with Applications to Energy Networks

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• 作者：Palle E. T. Jorgensen ; Erin P. J. Pearse
• 关键词：Graph energy ; Graph Laplacian ; Spectral graph theory ; Resistance network ; Effective resistance ; Hilbert space ; Reproducing kernel ; Unbounded linear operator ; Self ; adjoint extension ; Friedrichs extension ; Krein extension ; Essentially self ; adjoint ; Spectral resolution ; Defect indices ; Symmetric pair
• 刊名：Complex Analysis and Operator Theory
• 出版年：2016
• 出版时间：October 2016
• 年：2016
• 卷：10
• 期：7
• 页码：1535-1550
• 全文大小：542 KB
• 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
• 卷排序：10

文摘

We provide a streamlined construction of the Friedrichs extension of a densely-defined self-adjoint and semibounded operator A on a Hilbert space \(\mathcal {H}\), by means of a symmetric pair of operators. A symmetric pair is comprised of densely defined operators \(J: \mathcal {H}_1 \rightarrow \mathcal {H}_2\) and \(K: \mathcal {H}_2 \rightarrow \mathcal {H}_1\) which are compatible in a certain sense. With the appropriate definitions of \(\mathcal {H}_1\) and J in terms of A and \(\mathcal {H}\), we show that \((\textit{JJ}^\star )^{-1}\) is the Friedrichs extension of A. Furthermore, we use related ideas (including the notion of unbounded containment) to construct a generalization of the construction of the Krein extension of A as laid out in a previous paper of the authors. These results are applied to the study of the graph Laplacian on infinite networks, in relation to the Hilbert spaces \(\ell ^2(G)\) and \(\mathcal {H}_{\mathcal {E}}\) (the energy space).