Stability Properties Characterizing the Spectra of Operators on Banach Spaces

Let A ∈ L (E) be a contraction. The famous Katznelson-Tzafriri theorem [11, Theorem 1] states that the spectral condition σ (A) γ ⊆ {1} is equivalent to the convergence of the orbit {An(A − I): n = 1, 2, ...} in norm to zero. Assume that the orbit {An(A − I): n = 1, 2, ...} is relatively compact in L(E). Is there a spectral condition equivalent to this compactness? Such problems are studied for strongly continuous bounded representations of locally compact, abelian semigroups of linear operators on Banach spaces.