Spectral methods form a cornerstone of linear dynamics, where evolution is resolved into harmonic modes governed by eigenvalues and spectral measures of normal operators. For nonlinear dynamical systems, however, the harmonic eigenfunction paradigm typically breaks down: Koopman operators are often non-normal, may possess a continuous spectrum, and rarely admit complete eigenbases on natural observable spaces. This work develops a resolvent-centered operator-theoretic framework for generalized spectral representations of nonlinear flows through their associated Koopman C0 semigroups. Rather than relying on diagonalization, we construct resolvent-generated generalized spectral operators that yield weak integral representations of the semigroup valid in non-normal and continuous-spectrum regimes. We show that, under mild polynomial resolvent growth bounds along vertical lines, these spectral distributions become finite complex Radon measures on bounded spectral regions, thereby recovering a measure-theoretic interpretation analogous to classical spectral integrals. In the normal case, the framework reduces to the standard spectral theorem. The resulting resolvent-based perspective naturally incorporates pseudospectral amplification and transient growth, providing a unified description of both asymptotic and non-modal dynamics.