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We characterize linear operators preserving zero-restrictions on entire functions in weighted Bargmann-Fock spaces. This extends the characterization of linear operators on polynomials preserving stability (due to Borcea and the author) to the realm of entire functions, and translates into an optimal, albeit formal, Lee-Yang theorem.

The Lee-Yang and Pólya-Schur programs. III. Zero-preservers on Bargmann-Fock spaces