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.
"Definable Operators on Hilbert Spaces." Notre Dame J. Formal Logic 53 (2) 193 - 201, 2012. https://doi.org/10.1215/00294527-1715689