= Spectrum decomposition through the Banach-space adjoint
{title2=$\sigma(T)=\sigma_{\mathrm{ap}}(T)\cup\sigma_p(T')$}
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.
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