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Model theory of a Hilbert space expanded with an unbounded closed selfadjoint operator

We study a closed unbounded self‐adjoint operator Q acting on a Hilbert space H in the framework of Metric Abstract Elementary Classes (MAECs). We build a suitable MAEC for such a structure, prove it is ℵ0‐categorical and ℵ0‐stable up to a system of perturbations. We give an explicit continuous Lω1,ω axiomatization for the class. We also characterize non‐splitting and show it has the same properties as non‐forking in superstable first order theories. Finally, we characterize equality, orthogonality and domination of (Galois) types in that MAEC.