Unbounded operators in Hilbert space, duality rules, characteristic projections, and their applications

TL;DR

This paper develops a general framework for relating two Hilbert spaces via unbounded operators and duality, with applications to quantum physics, network models, and reflection positivity, without assuming boundedness.

Contribution

It introduces a new theorem linking two Hilbert spaces through a selfadjoint semibounded operator under natural assumptions, extending the theory of unbounded operators.

Findings

Established a comparison method between Hilbert spaces using a selfadjoint operator.

Applied the theory to physical Hamiltonians and infinite network models.

Provided insights into operator theory of reflection positivity.

Abstract

Our main theorem is in the generality of the axioms of Hilbert space, and the theory of unbounded operators. Consider two Hilbert spaces such that their intersection contains a fixed vector space D. It is of interest to make a precise linking between such two Hilbert spaces when it is assumed that D is dense in one of the two; but generally not in the other. 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 operator theory of reflection positivity.