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 ${\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s}}(T)$, 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 ${\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s}}(T)$ 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. ${\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s}}(T)={\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s}}(T+K)$. 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 $\{{\psi }_k{\}}_{k\in \mathbb{N}}\subseteq X$ in the Hilbert space $X$ such that $\Vert {\psi }_k\Vert =1$ and

$\underset{k\to \mathrm{\infty }}{\lim }\Vert (T-\lambda ){\psi }_k\Vert =0.$

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

The discrete spectrum

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

${\sigma }_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}(T)=\sigma (T)\setminus {\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s}}(T)$.

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

$\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}\{\psi \in X:T\psi =\lambda \psi \}$

has finite but non-zero dimension and that there is an $\epsilon >0$ such that $\mu \in \sigma (T)$ and $|\mu -\lambda |<\epsilon$ 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 ${\sigma }_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}(S)$ 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.

1. The essential spectrum ${\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s},1}(T)$ 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).
2. The essential spectrum ${\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s},2}(T)$ 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.
3. The essential spectrum ${\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s},3}(T)$ 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).
4. The essential spectrum ${\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s},4}(T)$ 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).
5. The essential spectrum ${\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s},5}(T)$ is the union of ${\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s},1}(T)$ with all components of $ℂ\setminus {\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s},1}(T)$ that do not intersect with the resolvent set $ℂ\setminus \sigma (T)$.

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

${\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s},1}(T)\subseteq {\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s},2}(T)\subseteq {\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s},3}(T)\subseteq {\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s},4}(T)\subseteq {\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s},5}(T)\subseteq \sigma (T)\subseteq ℂ,$

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

$r_{\mathrm{e}\mathrm{s}\mathrm{s},k}(T)=\max \{|\lambda |:\lambda \in {\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s},k}(T)\}.$

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

The essential spectrum ${\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s},k}(T)$ is invariant under compact perturbations for $k=1,2,3,4$, but not for $k=5$. The set ${\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s},4}(T)$ gives the part of the spectrum that is independent of compact perturbations, that is,

${\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s},4}(T)=\bigcap _{K\in B_0(X)}\sigma (T+K),$

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

$\sigma (T)={\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s},5}(T)⨆{\sigma }_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}(T)$,

where ${\sigma }_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}(T)$ is the discrete spectrum of $T$.

See also

• Spectrum (functional analysis)
• Resolvent formalism
• Decomposition of spectrum (functional analysis)
• Discrete spectrum (mathematics)
• Spectrum of an operator
• Operator theory
• Fredholm theory

References

Template:Reflist The self-adjoint case is discussed in

• Template:Citation
• Template:Cite book

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.