Work

Model Theory of Hilbert Spaces Expanded by Normal Operators

Year: 2026

Type: article

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 ∗ $T^*$ upper T Superscript asterisk and prove quantifier elimination for every completion after adding T ∗ $T^*$ upper T Superscript asterisk 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 $1$ 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 ω $\omega $ omega -stable up to perturbations.

Source: Bulletin of Symbolic Logic

DOI: 10.1017/bsl.2026.10176

Authors: ALEXANDER BERENSTEIN, NICOLAS CUERVO OVALLE, ISAAC GOLDBRING

Institutions: Universidad de Los Andes, Universidad de Los Andes, University of California, Irvine

Language: English

Topic: Advanced Topology and Set Theory

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,550

Funders: National Science Foundation, University of California, Irvine