Editing Spectrum (functional analysis) (section)

== Classification of points in the spectrum ==
{{further|Decomposition of spectrum (functional analysis)}}

A bounded operator ''T'' on a Banach space is invertible, i.e. has a bounded inverse, if and only if ''T'' is bounded below, i.e. <math>\|Tx\| \geq c\|x\|,</math> for some <math>c > 0,</math> and has dense range. Accordingly, the spectrum of ''T'' can be divided into the following parts:

# <math>\lambda\in\sigma(T)</math> if <math>T - \lambda I</math> is not bounded below. In particular, this is the case if <math>T - \lambda I</math> is not injective, that is, ''λ'' is an eigenvalue. The set of eigenvalues is called the '''point spectrum''' of ''T'' and denoted by ''σ''<sub>p</sub>(''T''). Alternatively, <math>T-\lambda I</math> could be one-to-one but still not bounded below. Such ''λ'' is not an eigenvalue but still an ''approximate eigenvalue'' of ''T'' (eigenvalues themselves are also approximate eigenvalues). The set of approximate eigenvalues (which includes the point spectrum) is called the '''approximate point spectrum''' of ''T'', denoted by ''σ''<sub>ap</sub>(''T'').
# <math>\lambda\in\sigma(T)</math> if <math>T-\lambda I</math> does not have dense range.  The set of such ''λ'' is called the '''compression spectrum''' of ''T'', denoted by <math>\sigma_{\mathrm{cp}}(T)</math>. If <math>T-\lambda I</math> does not have dense range but is injective, ''λ'' is said to be in the '''residual spectrum''' of ''T'', denoted by <math>\sigma_{\mathrm{r}}(T)</math>.

Note that the approximate point spectrum and residual spectrum are not necessarily disjoint<ref>{{Cite web |url=https://math.stackexchange.com/questions/1613668/nonempty-intersection-between-approximate-point-spectrum-and-residual-spectrum |title=Nonempty intersection between approximate point spectrum and residual spectrum}}</ref> (however, the point spectrum and the residual spectrum are).

The following subsections provide more details on the three parts of ''σ''(''T'') sketched above.

===Point spectrum===

If an operator is not injective (so there is some nonzero ''x'' with ''T''(''x'')&nbsp;=&nbsp;0), then it is clearly not invertible. So if ''λ'' is an [[eigenvalue]] of ''T'', one necessarily has ''λ''&nbsp;∈&nbsp;''σ''(''T''). The set of eigenvalues of ''T'' is also called the '''point spectrum''' of ''T'', denoted by ''σ''<sub>p</sub>(''T''). Some authors refer to the closure of the point spectrum as the '''pure point spectrum''' <math>\sigma_{pp}(T)=\overline{\sigma_{p}(T)}</math> while others simply consider <math>\sigma_{pp}(T):=\sigma_{p}(T).</math>{{sfn | Teschl | 2014 | p=115}}{{sfn|Simon|2005|page=44}}

===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.

===Discrete spectrum===

The [[discrete spectrum (mathematics)|discrete spectrum]] is defined as the set of [[normal eigenvalue]]s or, equivalently, as the set of isolated points of the spectrum such that the corresponding [[Riesz projector]] is of finite rank. As such, the discrete spectrum is a strict subset of the point spectrum, i.e., <math>\sigma_d(T) \subset \sigma_p(T).</math>

===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>.

===Compression spectrum===

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

===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>).

===Peripheral spectrum===

The peripheral spectrum of an operator is defined as the set of points in its spectrum which have modulus equal to its spectral radius.<ref name="Zaanen">{{cite book|last1=Zaanen|first1=Adriaan C.|title=Introduction to Operator Theory in Riesz Spaces|date=2012|publisher=Springer Science & Business Media|isbn=9783642606373|page=304|url=https://books.google.com/books?id=cgvpCAAAQBAJ&q=Peripheral+spectrum&pg=PA304|access-date=8 September 2017|language=en}}</ref>

===Essential spectrum===

There are five similar definitions of the [[essential spectrum]] of closed densely defined linear operator <math>A : \,X \to X </math> which satisfy

:<math>
\sigma_{\mathrm{ess},1}(A) \subset
\sigma_{\mathrm{ess},2}(A) \subset
\sigma_{\mathrm{ess},3}(A) \subset
\sigma_{\mathrm{ess},4}(A) \subset
\sigma_{\mathrm{ess},5}(A) \subset
\sigma(A).
</math>

All these spectra <math>\sigma_{\mathrm{ess},k}(A),\ 1\le k\le 5</math>, coincide in the case of self-adjoint operators.

# The essential spectrum <math>\sigma_{\mathrm{ess},1}(A)</math> is defined as the set of points <math>\lambda</math> of the spectrum such that <math>A-\lambda I</math> is not [[Fredholm operator|semi-Fredholm]]. (The operator is ''semi-Fredholm'' if its range is closed and either its kernel or cokernel (or both) is finite-dimensional.) <br>'''Example 1:''' <math>\lambda=0\in\sigma_{\mathrm{ess},1}(A)</math> for the operator <math>A:\,l^2(\N)\to l^2(\N)</math>, <math>A:\,e_j\mapsto e_j/j,~ j\in\N</math> (because the range of this operator is not closed: the range does not include all of <math>l^2(\N)</math> although its closure does).<br>'''Example 2:''' <math>\lambda=0\in\sigma_{\mathrm{ess},1}(N)</math> for <math>N:\,l^2(\N)\to l^2(\N)</math>, <math>N:\,v\mapsto 0</math> for any <math>v\in l^2(\N)</math> (because both kernel and cokernel of this operator are infinite-dimensional).
# The essential spectrum <math>\sigma_{\mathrm{ess},2}(A)</math> is defined as the set of points <math>\lambda</math> of the spectrum such that the operator either <math>A-\lambda I</math> has infinite-dimensional kernel or has a range which is not closed. It can also be characterized in terms of ''Weyl's criterion'': there exists a [[sequence]] <math>(x_j)_{j\in\N}</math> in the space ''X'' such that <math>\Vert x_j\Vert=1</math>, <math display="inline"> \lim_{j\to\infty} \left\|(A-\lambda I)x_j \right\| = 0,</math> and such that <math>(x_j)_{j\in\N}</math> contains no convergent [[subsequence]]. Such a sequence is called a ''singular sequence'' (or a ''singular Weyl sequence'').<br>'''Example:''' <math>\lambda=0\in\sigma_{\mathrm{ess},2}(B)</math> for the operator <math>B:\,l^2(\N)\to l^2(\N)</math>, <math>B:\,e_j\mapsto e_{j/2}</math> if ''j'' is even and <math>e_j\mapsto 0</math> when ''j'' is odd (kernel is infinite-dimensional; cokernel is zero-dimensional). Note that <math>\lambda=0\not\in\sigma_{\mathrm{ess},1}(B)</math>.
# The essential spectrum <math>\sigma_{\mathrm{ess},3}(A)</math> is defined as the set of points <math>\lambda</math> of the spectrum such that <math>A-\lambda I</math> is not [[Fredholm operator|Fredholm]]. (The operator is ''Fredholm'' if its range is closed and both its kernel and cokernel are finite-dimensional.) <br>'''Example:''' <math>\lambda=0\in\sigma_{\mathrm{ess},3}(J)</math> for the operator <math>J:\,l^2(\N)\to l^2(\N)</math>, <math>J:\,e_j\mapsto e_{2j}</math> (kernel is zero-dimensional, cokernel is infinite-dimensional). Note that <math>\lambda=0\not\in\sigma_{\mathrm{ess},2}(J)</math>.
# The essential spectrum <math>\sigma_{\mathrm{ess},4}(A)</math> is defined as the set of points <math>\lambda</math> of the spectrum such that <math>A-\lambda I</math> is not [[Fredholm operator|Fredholm]] of index zero. It could also be characterized as the largest part of the spectrum of ''A'' which is preserved by [[compact operator|compact]] perturbations. In other words, <math display="inline">\sigma_{\mathrm{ess},4}(A) = \bigcap_{K \in B_0(X)} \sigma(A+K)</math>; here <math>B_0(X)</math> denotes the set of all compact operators on ''X''. <br>'''Example:''' <math>\lambda=0\in\sigma_{\mathrm{ess},4}(R)</math> where <math>R:\,l^2(\N)\to l^2(\N)</math> is the right shift operator, <math>R:\,l^2(\N)\to l^2(\N)</math>, <math>R:\,e_j\mapsto e_{j+1}</math> for <math>j\in\N</math> (its kernel is zero, its cokernel is one-dimensional). Note that <math>\lambda=0\not\in\sigma_{\mathrm{ess},3}(R)</math>.
# The essential spectrum <math>\sigma_{\mathrm{ess},5}(A)</math> is the union of <math>\sigma_{\mathrm{ess},1}(A)</math> with all components of <math>\Complex \setminus \sigma_{\mathrm{ess},1}(A)</math> that do not intersect with the resolvent set <math>\Complex \setminus \sigma(A)</math>. It can also be characterized as <math>\sigma(A)\setminus\sigma_{\mathrm{d}}(A)</math>.<br>'''Example:''' consider the operator <math>T:\,l^2(\Z)\to l^2(\Z)</math>, <math>T:\,e_j\mapsto e_{j-1}</math> for <math>j\ne 0</math>, <math>T:\,e_0\mapsto 0</math>. Since <math>\Vert T\Vert=1</math>, one has <math>\sigma(T)\subset\overline{\mathbb{D}_1}</math>. For any <math>z\in\Complex</math> with <math>|z|=1</math>, the range of <math>T-z I</math> is dense but not closed, hence the boundary of the unit disc is in the first type of the essential spectrum: <math>\partial\mathbb{D}_1\subset\sigma_{\mathrm{ess},1}(T)</math>. For any <math>z\in\Complex</math> with <math>|z|<1</math>, <math>T-z I</math> has a closed range, one-dimensional kernel, and one-dimensional cokernel, so <math>z\in\sigma(T)</math> although <math>z\not\in\sigma_{\mathrm{ess},k}(T)</math> for <math>1\le k\le 4</math>; thus, <math>\sigma_{\mathrm{ess},k}(T)=\partial\mathbb{D}_1</math> for <math>1\le k\le 4</math>. There are two components of <math>\Complex\setminus\sigma_{\mathrm{ess},1}(T)</math>: <math>\{z\in\Complex:\,|z|>1\}</math> and <math>\{z\in\Complex:\,|z|<1\}</math>. The component <math>\{|z|<1\}</math> has no intersection with the resolvent set; by definition, <math>\sigma_{\mathrm{ess},5}(T)=\sigma_{\mathrm{ess},1}(T)\cup\{z\in\Complex:\,|z|<1\}=\{z\in\Complex:\,|z|\le 1\}</math>.