Editing Spectrum (functional analysis) (section)

===Approximate point spectrum===

More generally, by the [[bounded inverse theorem]], ''T'' is not invertible if it is not bounded below; that is, if there is no ''c''&nbsp;>&nbsp;0 such that ||''Tx''||&nbsp;≥&nbsp;''c''||''x''|| for all {{nowrap|''x'' ∈ ''X''}}.  So the spectrum includes the set of '''approximate eigenvalues''', which are those ''λ'' such that {{nowrap|''T'' - ''λI''}} is not bounded below; equivalently, it is the set of ''λ'' for which there is a sequence of unit vectors ''x''<sub>1</sub>, ''x''<sub>2</sub>, ... for which

:<math>\lim_{n \to \infty} \|Tx_n - \lambda x_n\| = 0</math>.

The set of approximate eigenvalues is known as the '''approximate point spectrum''', denoted by <math>\sigma_{\mathrm{ap}}(T)</math>.

It is easy to see that the eigenvalues lie in the approximate point spectrum.

For example, consider the right shift ''R'' on <math>l^2(\Z)</math> defined by

:<math>R:\,e_j\mapsto e_{j+1},\quad j\in\Z,</math>

where <math>\big(e_j\big)_{j\in\N}</math> is the standard orthonormal basis in <math>l^2(\Z)</math>. Direct calculation shows ''R'' has no eigenvalues, but every ''λ'' with <math>|\lambda|=1</math> is an approximate eigenvalue; letting ''x''<sub>''n''</sub> be the vector

:<math>\frac{1}{\sqrt{n}}(\dots, 0, 1, \lambda^{-1}, \lambda^{-2}, \dots, \lambda^{1 - n}, 0, \dots)</math>

one can see that ||''x''<sub>''n''</sub>|| = 1 for all ''n'', but

:<math>\|Rx_n - \lambda x_n\| = \sqrt{\frac{2}{n}} \to 0.</math>

Since ''R'' is a unitary operator, its spectrum lies on the unit circle. Therefore, the approximate point spectrum of ''R'' is its entire spectrum.

This conclusion is also true for a more general class of operators.
A unitary operator is [[normal operator|normal]]. By the [[spectral theorem]], a bounded operator on a Hilbert space H is normal if and only if it is equivalent (after identification of ''H'' with an <math>L^2</math> space) to a [[multiplication operator]]. It can be shown that the approximate point spectrum of a bounded multiplication operator equals its spectrum.