Comments on the cosmic convergence of nonexpansive maps

The paper’s Theorem 1.1/3.1 asserts: for a real Hilbert space H and a nonexpansive affine map T(x)=Ux+v with ||U||≤1, if 0 ∉ Im(I−T) (i.e., T has no fixed point), then there exists a nonzero continuous linear functional f with f(Tx) ≥ f(x) for all x; moreover ker f is a nontrivial closed U-invariant subspace. The proof appeals to Pazy/Gaubert–Vigeral to claim that the escape rate τ>0 and then selects a metric functional which, in a Hilbert space, must be linear, yielding the desired f (see the statement and proof snippets in Theorem 1.1/3.1) . However, the key implication “0 ∉ Im(I−T) ⇒ τ>0” is false in general. A standard counterexample is T(x)=Sx+e_1 on ℓ^2(ℕ), where S is the unilateral shift; here T has no fixed point (so 0 ∉ Im(I−T)), but τ=0 since ||T^n0||=√n. In this example Ker(I−S^*)={0}, so no nonzero f can satisfy f(Tx) ≥ f(x) ∀x; indeed, from the inequality one must have f∘U=f, i.e., U^*w=w for the representing vector w, which is impossible for S^* on ℓ^2. Thus Theorem 1.1 is false as stated. The proof’s use of τ>0 under 0 ∉ Im(I−T) is the source of the error; the correct condition is 0 ∉ cl(Im(I−T)), which would give τ>0 and a valid construction of f. The paper’s downstream argument that ker f is U-invariant is correct once such an f exists .