Editing Spectrum (functional analysis) (section)

===Continuous spectrum===

The set of all ''λ'' for which <math>T-\lambda I</math> is injective and has dense range, but is not surjective, is called the '''continuous spectrum''' of ''T'', denoted by  <math>\sigma_{\mathbb{c}}(T)</math>. The continuous spectrum therefore consists of those approximate eigenvalues which are not eigenvalues and do not lie in the residual spectrum. That is,

:<math>\sigma_{\mathrm{c}}(T) = \sigma_{\mathrm{ap}}(T) \setminus (\sigma_{\mathrm{r}}(T) \cup  \sigma_{\mathrm{p}}(T)) </math>.

For example, <math>A:\,l^2(\N)\to l^2(\N)</math>, <math>e_j\mapsto e_j/j</math>, <math>j\in\N</math>, is injective and has a dense range, yet <math>\mathrm{Ran}(A)\subsetneq l^2(\N)</math>.
Indeed, if <math display="inline">x = \sum_{j\in\N} c_j e_j\in l^2(\N)</math> with <math>c_j \in \Complex</math> such that <math display="inline">\sum_{j\in\N} |c_j|^2 < \infty</math>, one does not necessarily have <math display="inline">\sum_{j\in\N} \left|j c_j\right|^2 < \infty</math>, and then <math display="inline">\sum_{j\in\N} j c_j e_j \notin l^2(\N)</math>.