Resolvent-Based Stability Criterion for Non-Normal Dynamical Systems via the Kaupp Number

Abstract Classical stability theory relies on eigenvalue analysis of linear operators, assuming that negative real parts of eigenvalues guarantee stability. This criterion fails for non-normal systems, where transient amplification may occur despite asymptotic decay. We introduce a capacity-based stability criterion defined via the Kaupp number K = \frac{\|x(0)\| G_{\max}}{S}, \quad G_{\max} = \sup_{t \ge 0} \|e^{At}\|. We establish a resolvent-based lower bound linking transient amplification to pseudospectral structure and show that instability occurs when K \ge 1, even if all eigenvalues lie in the left half-plane.