Scalar characterization for strong and ordinary asymptotic expansivity

Find subsets a(T) and s(T) satisfying a(T)\subset s(T)\subset \tau(T) such that, for every scalar c, the operator cT is strong asymptotically expansive if and only if c\notin s(T), and asymptotically expansive if and only if c\notin a(T).

Background

For a complex Banach space operator T, the paper defines \tau(T) as the set of scalars c for which cT has spectrum meeting the unit circle. Proposition \ref{super_ct} completely characterizes the scalars producing super asymptotic expansivity, but only partial implications are obtained for strong and ordinary asymptotic expansivity. The unresolved problem is to identify subsets s(T) and a(T) that yield exact scalar characterizations for these two weaker notions.