Zero-product preserving additive maps on symmetric operator spaces and self-adjoint operator spaces

In this note, we characterize the additive maps on the space of symmetric operators and the space of self-adjoint operators which preserve zero-products in both directions, and the additive maps on the space of self-adjoint operators which preserve Jordan zero-products in both directions. We also give a complete classification of additive maps on the von Neumann algebra of all bounded linear operators acting on a Hilbert space which preserve square zero in both directions or preserve operators annihilated by a polynomial in both directions.