Spectrum decomposition through the Banach-space adjoint ($\sigma (T)={\sigma }_{\mathrm{ap}}(T)\cup {\sigma }_p(T^{\prime })$)

For a bounded linear operator on a complex Banach space, either a spectral scalar has approximate unit eigenvectors or it is an eigenvalue of the Banach-space adjoint. Indeed outside the approximate point spectrum, $S=T-\lambda I$ is bounded below, hence injective with closed range. If the adjoint has no zero vector in its kernel other than zero, the Hahn-Banach theorem makes the range dense: any proper closed range would have a nonzero continuous functional annihilating it. Thus the range is all of the space and the lower bound gives a bounded inverse. Conversely invertibility rules out both alternatives. With complex-linear duals there is no conjugation of $\lambda$ in the displayed formula; Hilbert-adjoint conventions must be distinguished.