Editing Spectrum (functional analysis) (section)

===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>.