Editing Spectrum (functional analysis) (section)

===Residual spectrum===

The set of <math>\lambda\in\Complex</math> for which <math>T-\lambda I</math> is injective but does not have dense range is known as the '''residual spectrum''' of ''T'' and is denoted by  <math>\sigma_{\mathrm{r}}(T)</math>:
:<math>\sigma_{\mathrm{r}}(T) = \sigma_{\mathrm{cp}}(T) \setminus \sigma_{\mathrm{p}}(T).</math>

An operator may be injective, even bounded below, but still not invertible. The right shift on <math>l^2(\mathbb{N})</math>, <math>R:\,l^2(\mathbb{N})\to l^2(\mathbb{N})</math>, <math>R:\,e_j\mapsto e_{j+1},\,j\in\N</math>, is such an example. This shift operator is an [[isometry]], therefore bounded below by 1. But it is not invertible as it is not surjective (<math>e_1\not\in\mathrm{Ran}(R)</math>), and moreover <math>\mathrm{Ran}(R)</math> is not dense in <math>l^2(\mathbb{N})</math>
(<math>e_1\notin\overline{\mathrm{Ran}(R)}</math>).