"A Unified Operator-Theoretic Framework for Stability in Non-Normal Systems: Synthesis and Outlook"

Abstract We present a unified operator-theoretic framework for stability in non-normal dynamical systems, synthesizing results on transient amplification, pseudospectral analysis, combinatorial interaction complexity, and control. Classical eigenvalue-based criteria are shown to be insufficient for predicting finite-time behavior. The framework centers on a dimensionless instability functional K(t) and its logarithmic growth rate \mathcal{T}(t), which together provide a complete description of system regimes and transitions. We integrate results across networked, continuum, and strongly coupled systems, and outline computational and control methodologies for stability assessment and mitigation.