Unbounded containment in the energy space of a network and the Krein extension of the energy Laplacian

2015/04/06 by Palle E. T. Jorgensen, Jorgensen, Palle E. T., Erin P. J. Pearse +1 · 1 citation
Mathematics · #05C75 #46E22 #46E22. Secondary: 05C50 #47B25 #FOS: Mathematics #Functional Analysis (math.FA) #Primary: 47B32 #math.FA #msc:05C50 #msc:05C75 #msc:46E22 #msc:46E22. #msc:47B25 #msc:47B32

paper · pdf · doi:10.48550/arxiv.1504.01332

17 pages, 3 figures. arXiv admin note: text overlap with arXiv:1103.5792

arxiv created 2015/07/10 · arxiv updated 2015/07/13

Abstract

We compare the space of square-summable functions on an infinite graph (denoted ℓ²(G)) with the space of functions of finite energy (denoted H_E). There is a notion of inclusion that allows ℓ²(G) to be embedded into H_E, but the required inclusion operator is unbounded in most interesting cases. These observations assist in the construction of the Krein extension of the Laplace operator on H_E. We investigate the Krein extension and compare it to the Friedrichs extension developed by the authors in a previous paper.

Citations

• Self-adjoint extensions of network Laplacians and applications to resistance metrics

Cited by

• Symmetric pairs and self-adjoint extensions of operators, with applications to energy networks

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