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 …

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 A∈M_n() with W(A)⊂Ω , on let R(λ)=q(λ)(h(λ)I-A)^ -1 . Then R(λ)=∑_ k≥0 b_k(A)λ^ -k , R(λ)+R(λ)^*0. For the fixed A,h , there are constants C<∞ and 0<θ<1 such that \|b_k(A)\|≤Cθ^k for all k≥0 . Thus the series converges absolutely and uniformly on . In particular, ∫_ R\,dτ=I and ∫_(R+R^*)\,dτ=2I .