The Schr\'oder-Bernstein property for operators on Hilbert spaces

We establish that the complete theory of a Hilbert space equipped with a normal operator has the Schr\"oder-Bernstein property for separable models. This is a partial answer to a question of Argoty, Berenstein, and the first-named author. We also prove an analogous statement for unbounded self-adjoint operators.
Comment: 6 pages; second draft. An error was discovered in the first version of the paper; the error has been repaired but now the result only holds for operators on separable Hilbert spaces. Comments still welcome!