Resolvent-Based Instability and the Kaupp Functional in Non-Normal Systems

Abstract We develop a rigorous operator-theoretic framework for instability in non-normal linear systems. Classical eigenvalue criteria fail to control transient amplification. We introduce the Kaupp functional K(t)=\frac{\|\Delta \mathcal{A}(t)\|}{d(\mathcal{A}(t))}\cdot \kappa(V)\cdot (1+\tau), and establish sufficient conditions for operational collapse. Using semigroup theory and resolvent estimates, we derive lower bounds for transient growth and connect them to pseudospectral geometry. We show that instability corresponds to a phase transition in operator space and provide computable criteria and examples.