Model Theory of a Hilbert Space Expanded with an Unbounded Closed Selfadjoint Operator

TL;DR

This paper develops a model-theoretic framework for Hilbert spaces expanded by unbounded selfadjoint operators, establishing stability and type properties within Metric Abstract Elementary Classes.

Contribution

It introduces a MAEC for Hilbert spaces with unbounded operators, proving stability and characterizing types, non-splitting, orthogonality, and domination.

Findings

MAEC for (H,Q) is aleph 0 stable up to perturbations

Characterization of non-splitting and non-forking properties

Types' orthogonality and domination are characterized

Abstract

We study a closed unbounded self-adoint operator Q acting on a Hilbert space H in the framework of Metric Abstract Elementary Classes (MAECS). We build a suitable MAEC for (H,Q), prove it is aleph 0 stable up to perturbations and characterize non-splitting and show it has the same properties as non-forking in superstable first order theorues. Also, we characterize equality, orthogonality and domination of (Galois) types in that MAEC.