Model theory of Hilbert spaces expanded by normal operators

TL;DR

This paper develops a model-theoretic framework for Hilbert spaces expanded by a bounded normal operator, establishing axioms, quantifier elimination, and stability properties linked to the operator's spectrum.

Contribution

It provides a complete axiomatization, quantifier elimination results, and a stability analysis for the theory of Hilbert spaces expanded by a normal operator, connecting types with spectral measures.

Findings

All completions are stable and characterized by the spectrum.

Quantifier elimination holds after adding the adjoint operator.

Types correspond to measures on the spectrum, with the logic topology matching the weak*-topology.

Abstract

We study expansions of Hilbert spaces with a bounded normal operator $T$. We axiomatize this theory in a natural language and identify all of its completions. We prove the definability of the adjoint $T^*$ and prove quantifier elimination for every completion after adding $T^*$ to the language. We identify types with measures on the spectrum of the operator and show that the logic topology on the type space corresponds to the weak*-topology on the space of measures. We also give a precise formula for the metric on the space of $1$-types. We prove all completions are stable and characterize the stability spectrum of the theory in terms of the spectrum of the operator. We also show all completions, regardless of their spectrum, are $\omega$-stable up to perturbations.