Koopman and Perron-Frobenius Operators on reproducing kernel Banach spaces

The paper’s Theorem 4.7 (items 1–8) and Lemma 4.5 establish exactly the adjoint relation Uf = K′f on the algebraic core, functoriality, invertibility for bijections, identifiability, closedness/boundedness of Uf, non-closedness of Kf under the universal property, the equivalence between dense domain and closability (under φ-isomorphism and reflexivity), and the measure-based extension of Kf; the candidate solution reproduces these arguments point for point, using the same core identities and measure-approximation device. The only minor gaps are (i) not stating f’s continuity explicitly in item (5) though it is used implicitly, and (ii) asserting h ∉ Span{k(x,·)} from “μ has infinite support” without invoking the paper’s uniqueness-of-representation argument via the universal property; both are easily fixed and are handled in the paper’s proofs (see Theorem 4.7 and its proof, especially the measure extension and uniqueness arguments).