Essential spectrum

In mathematics, the essential spectrum of a bounded operator (or, more generally, of a densely defined closed linear operator) is a certain subset of its spectrum, defined by a condition of the type that says, roughly speaking, "fails badly to be invertible".

The essential spectrum of self-adjoint operatorsEdit

In formal terms, let <math>X</math> be a Hilbert space and let <math>T</math> be a self-adjoint operator on <math>X</math>.

DefinitionEdit

The essential spectrum of <math>T</math>, usually denoted <math>\sigma_{\mathrm{ess}}(T)</math>, is the set of all real numbers <math>\lambda \in \R</math> such that

<math>T-\lambda I_X</math>

is not a Fredholm operator, where <math>I_X</math> denotes the identity operator on <math>X</math>, so that <math>I_X(x)=x</math>, for all <math>x \in X</math>. (An operator is Fredholm if its kernel and cokernel are finite-dimensional.)

The definition of essential spectrum <math>\sigma_{\mathrm{ess}}(T)</math> will remain unchanged if we allow it to consist of all those complex numbers <math>\lambda \in \C</math> (instead of just real numbers) such that the above condition holds. This is due to the fact that the spectrum of self-adjoint consists only of real numbers.

PropertiesEdit

The essential spectrum is always closed, and it is a subset of the spectrum <math>\sigma(T)</math>. As mentioned above, since <math>T</math> is self-adjoint, the spectrum is contained on the real axis.

The essential spectrum is invariant under compact perturbations. That is, if <math>K</math> is a compact self-adjoint operator on <math>X</math>, then the essential spectra of <math>T</math> and that of <math>T+K</math> coincide, i.e. <math>\sigma_{\mathrm{ess}}(T)=\sigma_{\mathrm{ess}}(T+K)</math>. This explains why it is called the essential spectrum: Weyl (1910) originally defined the essential spectrum of a certain differential operator to be the spectrum independent of boundary conditions.

Weyl's criterion is as follows. First, a number <math>\lambda</math> is in the spectrum <math>\sigma(T)</math> of the operator <math>T</math> if and only if there exists a sequence <math>\{\psi_k\}_{k\in \N} \subseteq X</math> in the Hilbert space <math>X</math> such that <math>\Vert\psi_k\Vert=1</math> and

<math> \lim_{k\to\infty} \left\| (T - \lambda)\psi_k \right\| = 0. </math>

Furthermore, <math>\lambda</math> is in the essential spectrum if there is a sequence satisfying this condition, but such that it contains no convergent subsequence (this is the case if, for example <math>\{\psi_k\}_{k\in \N}</math> is an orthonormal sequence); such a sequence is called a singular sequence. Equivalently, <math>\lambda</math> is in the essential spectrum <math>\sigma_{\mathrm{ess}}(T)</math> if there exists a sequence satisfying the above condition, which also converges weakly to the zero vector <math>\mathbf{0}_X</math> in <math>X</math>.

The discrete spectrumEdit

The essential spectrum <math>\sigma_{\mathrm{ess}}(T)</math> is a subset of the spectrum <math>\sigma(T)</math> and its complement is called the discrete spectrum, so

<math> \sigma_{\mathrm{disc}}(T) = \sigma(T) \setminus \sigma_{\mathrm{ess}}(T)</math>.

If <math>T</math> is self-adjoint, then, by definition, a number <math>\lambda</math> is in the discrete spectrum <math>\sigma_{\mathrm{disc}}</math> of <math>T</math> if it is an isolated eigenvalue of finite multiplicity, meaning that the dimension of the space

<math> \ \mathrm{span} \{ \psi \in X : T\psi = \lambda\psi \} </math>

has finite but non-zero dimension and that there is an <math>\varepsilon>0</math> such that <math>\mu \in \sigma(T)</math> and <math>|\mu-\lambda|<\varepsilon</math> imply that <math>\mu</math> and <math>\lambda</math> are equal. (For general, non-self-adjoint operators <math>S</math> on Banach spaces, by definition, a complex number <math>\lambda \in \C</math> is in the discrete spectrum <math>\sigma_{\mathrm{disc}}(S)</math> if it is a normal eigenvalue; or, equivalently, if it is an isolated point of the spectrum and the rank of the corresponding Riesz projector is finite.)

The essential spectrum of closed operators in Banach spacesEdit

Let <math>X</math> be a Banach space and let <math>T:\,D(T)\to X</math> be a closed linear operator on <math>X</math> with dense domain <math>D(T)</math>. There are several definitions of the essential spectrum, which are not equivalent.<ref>Template:Cite journal</ref>

The essential spectrum <math>\sigma_{\mathrm{ess},1}(T)</math> is the set of all <math>\lambda</math> such that <math>T-\lambda I_X</math> is not semi-Fredholm (an operator is semi-Fredholm if its range is closed and its kernel or its cokernel is finite-dimensional).
The essential spectrum <math>\sigma_{\mathrm{ess},2}(T)</math> is the set of all <math>\lambda</math> such that the range of <math>T-\lambda I_X</math> is not closed or the kernel of <math>T-\lambda I_X</math> is infinite-dimensional.
The essential spectrum <math>\sigma_{\mathrm{ess},3}(T)</math> is the set of all <math>\lambda</math> such that <math>T-\lambda I_X</math> is not Fredholm (an operator is Fredholm if its range is closed and both its kernel and its cokernel are finite-dimensional).
The essential spectrum <math>\sigma_{\mathrm{ess},4}(T)</math> is the set of all <math>\lambda</math> such that <math>T-\lambda I_X</math> is not Fredholm with index zero (the index of a Fredholm operator is the difference between the dimension of the kernel and the dimension of the cokernel).
The essential spectrum <math>\sigma_{\mathrm{ess},5}(T)</math> is the union of <math>\sigma_{\mathrm{ess},1}(T)</math> with all components of <math>\C\setminus \sigma_{\mathrm{ess},1}(T)</math> that do not intersect with the resolvent set <math>\C \setminus \sigma(T)</math>.

Each of the above-defined essential spectra <math>\sigma_{\mathrm{ess},k}(T)</math>, <math>1\le k\le 5</math>, is closed. Furthermore,

<math> \sigma_{\mathrm{ess},1}(T) \subseteq \sigma_{\mathrm{ess},2}(T) \subseteq \sigma_{\mathrm{ess},3}(T) \subseteq \sigma_{\mathrm{ess},4}(T) \subseteq \sigma_{\mathrm{ess},5}(T) \subseteq \sigma(T) \subseteq \C,</math>

and any of these inclusions may be strict. For self-adjoint operators, all the above definitions of the essential spectrum coincide.

Define the radius of the essential spectrum by

<math>r_{\mathrm{ess},k}(T) = \max \{ |\lambda| : \lambda\in\sigma_{\mathrm{ess},k}(T) \}. </math>

Even though the spectra may be different, the radius is the same for all <math>k=1,2,3,4,5</math>.

The definition of the set <math>\sigma_{\mathrm{ess},2}(T)</math> is equivalent to Weyl's criterion: <math>\sigma_{\mathrm{ess},2}(T)</math> is the set of all <math>\lambda</math> for which there exists a singular sequence.

The essential spectrum <math>\sigma_{\mathrm{ess},k}(T)</math> is invariant under compact perturbations for <math>k=1,2,3,4</math>, but not for <math>k=5</math>. The set <math>\sigma_{\mathrm{ess},4}(T)</math> gives the part of the spectrum that is independent of compact perturbations, that is,

<math> \sigma_{\mathrm{ess},4}(T) = \bigcap_{K \in B_0(X)} \sigma(T+K), </math>

where <math>B_0(X)</math> denotes the set of compact operators on <math>X</math> (D.E. Edmunds and W.D. Evans, 1987).

The spectrum of a closed, densely defined operator <math>T</math> can be decomposed into a disjoint union

<math>\sigma(T)=\sigma_{\mathrm{ess},5}(T)\bigsqcup\sigma_{\mathrm{disc}}(T)</math>,

where <math>\sigma_{\mathrm{disc}}(T)</math> is the discrete spectrum of <math>T</math>.

ReferencesEdit

Template:Reflist The self-adjoint case is discussed in

A discussion of the spectrum for general operators can be found in

D.E. Edmunds and W.D. Evans (1987), Spectral theory and differential operators, Oxford University Press. Template:ISBN.

The original definition of the essential spectrum goes back to

H. Weyl (1910), Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen, Mathematische Annalen 68, 220–269.

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