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 operators

In formal terms, let $X$ be a Hilbert space and let $T$ be a self-adjoint operator on $X$ .

=Definition=

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

: $T-\lambda I_X$

is not a Fredholm operator, where $I_X$ denotes the identity operator on $X$ , so that $I_X(x)=x$ , for all $x \in X$ .

(An operator is Fredholm if its kernel and cokernel are finite-dimensional.)

The definition of essential spectrum $\sigma_{\mathrm{ess}}(T)$ will remain unchanged if we allow it to consist of all those complex numbers $\lambda \in \C$ (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.

=Properties=

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

The essential spectrum is invariant under compact perturbations. That is, if $K$ is a compact self-adjoint operator on $X$ , then the essential spectra of $T$ and that of $T+K$ coincide, i.e. $\sigma_{\mathrm{ess}}(T)=\sigma_{\mathrm{ess}}(T+K)$ . 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 $\lambda$ is in the spectrum $\sigma(T)$ of the operator $T$ if and only if there exists a sequence $\{\psi_k\}_{k\in \N} \subseteq X$ in the Hilbert space $X$ such that $\Vert\psi_k\Vert=1$ and

: $\lim_{k\to\infty} \left\| (T - \lambda)\psi_k \right\| = 0.$

Furthermore, $\lambda$ 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 $\{\psi_k\}_{k\in \N}$ is an orthonormal sequence); such a sequence is called a singular sequence. Equivalently, $\lambda$ is in the essential spectrum $\sigma_{\mathrm{ess}}(T)$ if there exists a sequence satisfying the above condition, which also converges weakly to the zero vector $\mathbf{0}_X$ in $X$ .

=The discrete spectrum=

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

: $\sigma_{\mathrm{disc}}(T) = \sigma(T) \setminus \sigma_{\mathrm{ess}}(T)$ .

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

: $\ \mathrm{span} \{ \psi \in X : T\psi = \lambda\psi \}$

has finite but non-zero dimension and that there is an $\varepsilon>0$ such that $\mu \in \sigma(T)$ and $|\mu-\lambda|<\varepsilon$ imply that $\mu$ and $\lambda$ are equal.

(For general, non-self-adjoint operators $S$ on Banach spaces, by definition, a complex number $\lambda \in \C$ is in the discrete spectrum $\sigma_{\mathrm{disc}}(S)$ 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 spaces

Let $X$ be a Banach space

and let $T:\,D(T)\to X$ be a closed linear operator on $X$ with dense domain $D(T)$ . There are several definitions of the essential spectrum, which are not equivalent.{{cite journal |last1=Gustafson |first1=Karl|author-link=Karl Edwin Gustafson |title=On the essential spectrum |journal=Journal of Mathematical Analysis and Applications |date=1969 |volume=25 |issue=1 |pages=121–127 |url=https://core.ac.uk/download/pdf/82202846.pdf}}

The essential spectrum $\sigma_{\mathrm{ess},1}(T)$ is the set of all $\lambda$ such that $T-\lambda I_X$ 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 $\sigma_{\mathrm{ess},2}(T)$ is the set of all $\lambda$ such that the range of $T-\lambda I_X$ is not closed or the kernel of $T-\lambda I_X$ is infinite-dimensional.
The essential spectrum $\sigma_{\mathrm{ess},3}(T)$ is the set of all $\lambda$ such that $T-\lambda I_X$ 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 $\sigma_{\mathrm{ess},4}(T)$ is the set of all $\lambda$ such that $T-\lambda I_X$ 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 $\sigma_{\mathrm{ess},5}(T)$ is the union of $\sigma_{\mathrm{ess},1}(T)$ with all components of $\C\setminus \sigma_{\mathrm{ess},1}(T)$ that do not intersect with the resolvent set $\C \setminus \sigma(T)$ .

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

: $\sigma_{\mathrm{ess},1}(T) \subseteq \sigma_{\mathrm{ess},2}(T) \subseteq \sigma_{\mathrm{ess},3}(T) \subseteq \sigma_{\mathrm{ess},4}(T) \subseteq \sigma_{\mathrm{ess},5}(T) \subseteq \sigma(T) \subseteq \C,$

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

: $r_{\mathrm{ess},k}(T) = \max \{ |\lambda| : \lambda\in\sigma_{\mathrm{ess},k}(T) \}.$

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

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

The essential spectrum $\sigma_{\mathrm{ess},k}(T)$ is invariant under compact perturbations for $k=1,2,3,4$ , but not for $k=5$ .

The set $\sigma_{\mathrm{ess},4}(T)$ gives the part of the spectrum that is independent of compact perturbations, that is,

: $\sigma_{\mathrm{ess},4}(T) = \bigcap_{K \in B_0(X)} \sigma(T+K),$

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

The spectrum of a closed, densely defined operator $T$ can be decomposed into a disjoint union

: $\sigma(T)=\sigma_{\mathrm{ess},5}(T)\bigsqcup\sigma_{\mathrm{disc}}(T)$ ,

where $\sigma_{\mathrm{disc}}(T)$ is the discrete spectrum of $T$ .

References

The self-adjoint case is discussed in

{{citation |first1 = Michael C. |last1 = Reed|author1-link=Michael C. Reed |first2 = Barry |last2 = Simon |author2-link = Barry Simon |title = Methods of modern mathematical physics: Functional Analysis |volume=1 |place = San Diego |publisher = Academic Press |year = 1980 |isbn = 0-12-585050-6}}
{{Cite book |title=Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators |first= Gerald |last=Teschl |authorlink= Gerald Teschl |publisher= American Mathematical Society |year=2009 |url=https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/ |isbn=978-0-8218-4660-5 }}

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. {{ISBN|0-19-853542-2}}.

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.

Category:Spectral theory