[Paper Review] DUALITY FOR UNBOUNDED OPERATORS, AND APPLICATIONS

Palle E. T. Jørgensen, Feng Tian|arXiv (Cornell University)|Sep 26, 2015

Spectral Theory in Mathematical Physics11 references3 citations

TL;DR

This paper establishes a duality framework for unbounded operators in Hilbert spaces without assuming relative boundedness, linking two Hilbert spaces via a selfadjoint semibounded operator when a common dense subspace D exists in one but not necessarily the other. The key contribution is a general duality theorem with applications to physical Hamiltonians and reflection positivity in operator theory.

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.

Motivation & Objective

• To develop a duality theory for unbounded operators in Hilbert spaces under minimal assumptions.
• To address the case where a common dense subspace D exists in one Hilbert space but not in the other.
• To establish a comparison between two Hilbert spaces using a specific selfadjoint semibounded operator.
• To apply the duality framework to physical systems, including continuous and discrete Hamiltonians.
• To extend the theory of reflection positivity to unbounded settings using this duality.

Proposed method

• Formulate the duality using two Hilbert spaces sharing a common dense subspace D in one but not necessarily in the other.
• Introduce a selfadjoint semibounded operator as the central linking mechanism between the two Hilbert spaces.
• Use axioms of Hilbert space theory and structural assumptions from potential theory to ensure the duality holds.
• Avoid assumptions of relative boundedness, allowing broader applicability to unbounded operators.
• Construct a comparison map between the two Hilbert spaces via the spectral properties of the selfadjoint operator.
• Apply the framework to infinite network models and quantum mechanical Hamiltonians to validate the duality.

Experimental results

Research questions

• RQ1 How can two Hilbert spaces be linked when a common subspace D is dense in only one?
• RQ2 What conditions ensure a meaningful comparison between unbounded operators in different Hilbert spaces without relative boundedness?
• RQ3 Can a selfadjoint semibounded operator serve as a bridge between two Hilbert spaces with asymmetric density properties?
• RQ4 How does this duality framework extend to physical models such as continuous and discrete Hamiltonians?
• RQ5 In what way does this duality support the theory of reflection positivity for unbounded operators?

Key findings

• A duality theorem is established between two Hilbert spaces sharing a common subspace D, where D is dense in one but not necessarily in the other.
• The linking mechanism is a selfadjoint semibounded operator, which enables comparison despite lack of relative boundedness.
• The framework applies to both continuous and discrete physical Hamiltonians, including infinite network models.
• The theory supports reflection positivity in unbounded operator settings, extending its domain beyond bounded cases.
• The results are derived under natural assumptions motivated by potential theory, ensuring broad applicability.