Embeddings, operator ranges, and Dirac operators
Date
2011Source Title
Complex Analysis and Operator Theory
Print ISSN
1661-8254
Electronic ISSN
1661-8262
Publisher
Springer Basel
Volume
5
Issue
2
Pages
941 - 953
Language
English
Type
ArticleItem Usage Stats
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Abstract
Motivated by energy space representation of Dirac operators, in the sense of K. Friedrichs, we recently introduced the notion of closely embedded Kreǐn spaces. These spaces are associated to unbounded selfadjoint operators that play the role of kernel operators, in the sense of L. Schwartz, and they are special representations of induced Kreǐn spaces. In this article we present a canonical representation of closely embedded Kreǐn spaces in terms of a generalization of the notion of operator range and obtain a characterization of uniqueness. When applied to Dirac operators, the results differ according to a mass or a massless particle in a dramatic way: in the case of a particle with a nontrivial mass we obtain a dual of a Sobolev type space and we have uniqueness, while in the case of a massless particle we obtain a dual of a homogenous Sobolev type space and we lose uniqueness. © 2010 Elsevier Inc.
Keywords
Closed embeddingDirac operator
Homogenous Sobolev space
Kernel operator
Krein space
Operator range