Reflexive, Hilbert-space, and analytic-semigroup validity of the failed implications

Determine whether any of the implications that fail for general strongly continuous semigroups on non-reflexive Banach spaces become valid when the state space is reflexive or Hilbert, or when the semigroup is analytic.

Background

The paper establishes a hierarchy connecting exponential detectability, dissipation- and implication-form eOSS Lyapunov functions, eOSS, and exponential zero-detectability. Several converse implications fail in general, with counterexamples involving strongly continuous semigroups on non-reflexive Banach spaces. The authors explicitly leave unresolved whether these counterexamples persist, or whether some of the failed implications are restored, under stronger structural assumptions such as reflexivity, a Hilbert state space, or analyticity of the semigroup.