Non-Hermitian Resolvent Geometry: Pseudospectral Curvature and the Gauss-Bonnet-Schur Theorem

The classical spectral theorem fails for non-normal operators (A A† ≠ A† A), rendering isolated eigenvalues blind to pseudospectral instabilities, eigenvector non-orthogonality, and extreme perturbation sensitivities in open quantum systems and advective dynamics. We establish a rigorous geometric foundation to overcome this barrier by defining the Non-Hermitian Resolvent Metric g_A(z) = ||(A - zI)^(-1)||_F^2 (dx^2 + dy^2) on the complex resolvent set Ω = C \ σ(A). We prove that the Riemannian curvature associated with g_A(z) encodes the non-normality of A: while normal operators correspond to strictly flat resolvent manifolds (K_A ≡ 0), non-normal operators induce singular curvature wells K_A(z) → -∞ near pseudospectral boundaries. Our main result is a Gauss-Bonnet-type index theorem for non-normal operators. We prove that the integrated Gaussian curvature over an ε-pseudospectrum Λ_ε(A) obeys a topological identity: ∬{Λ_ε(A)} K_A(z) dA{g_A} + ∫{∂Λ_ε(A)} k_g ds{g_A} = 2π ( χ(Λ_ε(A)) - rank([A, A†]) ) where χ(Λ_ε(A)) is the Euler-Poincaré characteristic and rank([A, A†]) acts as a quantized defect charge arising from the Schur/Jordan non-normality. Furthermore, we demonstrate that the geodesic paths under g_A(z) minimize the operator norm of destabilizing perturbation trajectories E(t). This framework unites non-normal operator theory, differential geometry, and pseudospectral analysis into a unified, coordinate-invariant field theory of operator stability.