Entropy for compact operators and results on entropy and specification.

The paper proves that for a compact operator T on a Banach space the topological entropy equals the sum of log|λ| over eigenvalues with |λ|>1 (counted with multiplicity), via a Riesz decomposition and a direct appeal to Walters’s finite-dimensional entropy formula. The model independently derives the same formula by (i) extracting the finite-dimensional expanding subspace using Riesz–Schauder projections, (ii) proving zero entropy on the nonexpanding complement, and (iii) computing the finite-dimensional contribution by a uniform-expansion/volume-growth argument. The paper’s proof has a small gap where Lemma 3.2 is cited to conclude htop(T1)=0 even when σ(T1) may contain |λ|=1; the model correctly handles the central part (|λ|=1) to close this gap. Aside from this fixable point and a need to state multiplicity conventions explicitly, both arguments are correct and reach the same result.