Theorem 3.1 ([The resolvent expansion and double-layer positivity]). Let Ω,h,q , and (b_k) satisfy the hypotheses and conclusions of Theorem 2.1 ([Exterior Faber coefficients and the positive kernel]). For any positive integer n and …

Every eigenvalue of A belongs to W(A) , by testing a unit eigenvector. Thus the exterior resolvent is defined on and outside the boundary. Compactness of the circle and strict spectral separation allow a smaller collar |λ|>r_A , with r<r_A<1 , on which it remains holomorphic, including infinity where R(∞)=I . Expansion at infinity gives the coefficients b_k(A) : this follows either by the geometric resolvent series for large |h(λ)| or by substituting A into the polynomial coefficient identities in. Choose r_A<θ<1 . Cauchy estimates on the reciprocal circle of radius θ^ -1 give the stated geometric bound. They imply absolute uniform convergence on , and its constant coefficient b_0(A)=I gives both integrals. For B=h(λ)I-A , direct congruence gives B^*(R+R^*)B=q(λ)B+q(λ)B^*. For each unit vector v , the quadratic form on the right is 2(q(λ)(h(λ)-v^*Av)) , which is nonnegative by. Invertibility of B proves positivity.