Unbounded operators in Hilbert space, duality rules, characteristic projections, and their applications. (English) Zbl 1511.47030
Summary: Our main theorem is in the generality of the axioms of Hilbert space, and the theory of unbounded operators. Consider two Hilbert spaces whose intersection contains a fixed vector space \(\mathscr {D}\). In the case when \(\mathscr {D}\) is dense in one of the Hilbert spaces (but not necessarily in the other), we make precise an operator-theoretic linking between the two Hilbert spaces. No relative boundedness is assumed. Nonetheless, under natural assumptions (motivated by potential theory), we prove a theorem where a comparison between the two Hilbert spaces is made via a specific selfadjoint semibounded operator. Applications include physical Hamiltonians, both continuous and discrete (infinite network models), and the operator theory of reflection positivity.
MSC:
| 47B25 | Linear symmetric and selfadjoint operators (unbounded) |
| 46C05 | Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product) |
| 81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
Keywords:
quantum mechanics; unbounded operator; closable operator; selfadjoint extensions; spectral theory; reproducing kernel Hilbert space; discrete analysis; graph Laplacians; distribution of point-masses; Green’s functionsReferences:
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