Point continuous and residual spectrum

Let $X$ be a nonzero complex Banach space, let $T\in \mathcal{B}(X)$ be a bounded operator (A bounded linear operator between normed spaces) and let $\lambda \in \mathbb{C}$. Write $T-\lambda$ for $T-\lambda I_X$, where $I_X$ is the identity, and recall that $\lambda \in \sigma (T)$ exactly when $T-\lambda$ is not invertible (Spectrum and resolvent set in a Banach algebra).

The spectral value $\lambda$ is

• in the point spectrum ${\sigma }_p(T)$ when $T-\lambda$ is not injective, that is, when $\lambda$ is an eigenvalue;
• in the continuous spectrum ${\sigma }_c(T)$ when $T-\lambda$ is injective, has dense range, and is not surjective;
• in the residual spectrum ${\sigma }_r(T)$ when $T-\lambda$ is injective and its range is not dense in $X$.

The three sets are pairwise disjoint by their injectivity and density conditions, and each is contained in $\sigma (T)$: every listed condition precludes a two-sided inverse in $\mathcal{B}(X)$.

Assume additionally Dependent Choice (The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain) for the partition assertion. If $T-\lambda$ is injective and surjective, the bounded inverse theorem Bounded inverse theorem makes its inverse bounded. Consequently, for a spectral value with injective dense range, surjectivity is impossible. Splitting first by injectivity and then by density therefore gives ${\sigma }_p(T)\bigsqcup {\sigma }_c(T)\bigsqcup {\sigma }_r(T)=\sigma (T)\text{under DC}\mathrm{.}$ The definitions themselves do not require DC. For $T=0$ on the nonzero space, ${\sigma }_p(T)=\{0\}$ and the other two parts are empty, since $-\lambda I$ has inverse $-{\lambda }^{-1}I$ for $\lambda \mathrm{\ne }0$.

• The residual spectrum is the injective case with non-dense range. No closedness is presupposed: $\lambda \in {\sigma }_r(T)$ simply means that $T-\lambda$ is injective and its range is not dense in $X$. The reader should not add the hypothesis that the range be closed; the point of the definition is to separate dense range from non-dense range, and a non-dense range may still fail to be closed.

• Eigenvalues belong only to the point spectrum. If $T-\lambda$ is not injective, then $\lambda \in {\sigma }_p(T)$ and, whatever the range is, $\lambda$ is not in ${\sigma }_c(T)$ or ${\sigma }_r(T)$ by the disjoint classification above. Consequently ${\sigma }_r={\sigma }_{cp}\setminus {\sigma }_p$: after compression values that are eigenvalues are removed, the remaining operators are exactly the injective ones with non-dense range. This identity is recorded and proved in Relations among the five spectral parts rather than identifying ${\sigma }_r$ with the whole compression spectrum.

• Individual parts need not be closed. The spectrum is closed, but the point spectrum need not be, and the closures of the three disjoint parts may meet at accumulation points of the whole spectrum; under DC their union is the closed spectrum by the partition argument above.