Off-diagonal terms in symmetric operators

TL;DR

This paper compares classical and recent obstructions to symmetric operators being selfadjoint, introducing the concept of smooth projections and their role in spectral analysis.

Contribution

It introduces a new obstruction based on smooth projections for symmetric operators and compares it with the classical deficiency space approach.

Findings

Existence of smooth projections implies selfadjoint closure of the operator.

Smooth projections can diagonalize the operator, leading to spectral resolution.

The paper provides results for both single operators and systems of operators.

Abstract

In this paper we provide a quantitative comparison of two obstructions for a given symmetric operator S with dense domain in Hilbert space ${\cal H}$ to be selfadjoint. The first one is the pair of deficiency spaces of von Neumann, and the second one is of more recent vintage: Let P be a projection in ${\cal H}$. We say that it is smooth relative to S if its range is contained in the domain of S. We say that smooth projections $\{P_i \}_{i=1}^{\infty}$ diagonalize S if (a) $(I-P_{i})SP_i=0$ for all i, and (b) $\sup_{i}P_{i}=I$. If such projections exist, then S has a selfadjoint closure (i.e., $\bar{S}$ has a spectral resolution), and so our second obstruction to selfadjointness is defined from smooth projections $P_i$ with $(I-P_i)SP_i \neq 0$. We prove results both in the case of a single operator S and a system of operators.