Skew-adjointness of the orthogonal generator for all observables in D(L)

Determine whether, for a skew-adjoint generator L and the Mori projection operator P onto span(z) (with Q = I − P), the closure of Q L Q is skew-adjoint for every observable z in the domain D(L), not only for observables z in the dense subset F(L) where polynomial bounds on powers of L hold.

Background

Lemma 4.6 proves that if L is skew-adjoint and z lies in the dense subset F(L), then the closure of Q L Q is skew-adjoint. This result ensures favorable semigroup properties for the orthogonal dynamics in that setting.

The authors note it is unknown whether the same conclusion extends to all observables z in the full operator domain D(L). Establishing this would clarify the generality of the semigroup and unitary properties of the orthogonal dynamics under the Mori projection in equilibrium settings.