Definable Operators on Hilbert Spaces

TL;DR

This paper characterizes the linear operators on infinite-dimensional Hilbert spaces that are definable within the natural metric structure, showing they are precisely those operators that can be expressed as a scalar plus a compact operator.

Contribution

It provides a complete characterization of definable operators on Hilbert spaces, linking model-theoretic definability with operator-theoretic properties.

Findings

Definable operators are exactly scalar plus compact operators.

The characterization applies to both real and complex Hilbert spaces.

Provides a bridge between model theory and operator theory.

Abstract

Let H be an infinite-dimensional (real or complex) Hilbert space, viewed as a metric structure in its natural signature. We characterize the definable linear operators on H as exactly the "scalar plus compact" operators.