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 ${\displaystyle X}$ be a Hilbert space and let ${\displaystyle T}$ be a self-adjoint operator on ${\displaystyle X}$.

Definition

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

${\displaystyle T-\lambda I_{X}}$

is not a Fredholm operator, where ${\displaystyle I_{X}}$ denotes the identity operator on ${\displaystyle X}$, so that ${\displaystyle I_{X}(x)=x}$, for all ${\displaystyle x\in X}$. (An operator is Fredholm if its kernel and cokernel are finite-dimensional.)

The definition of essential spectrum ${\displaystyle \sigma _{\mathrm {ess} }(T)}$ will remain unchanged if we allow it to consist of all those complex numbers ${\displaystyle \lambda \in \mathbb {C} }$ (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 ${\displaystyle \sigma (T)}$. As mentioned above, since ${\displaystyle T}$ is self-adjoint, the spectrum is contained on the real axis.

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

${\displaystyle \lim _{k\to \infty }\left\|(T-\lambda )\psi _{k}\right\|=0.}$

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

The discrete spectrum

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

${\displaystyle \sigma _{\mathrm {disc} }(T)=\sigma (T)\setminus \sigma _{\mathrm {ess} }(T)}$.

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

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

has finite but non-zero dimension and that there is an ${\displaystyle \varepsilon >0}$ such that ${\displaystyle \mu \in \sigma (T)}$ and ${\displaystyle |\mu -\lambda |<\varepsilon }$ imply that ${\displaystyle \mu }$ and ${\displaystyle \lambda }$ are equal. (For general, non-self-adjoint operators ${\displaystyle S}$ on Banach spaces, by definition, a complex number ${\displaystyle \lambda \in \mathbb {C} }$ is in the discrete spectrum ${\displaystyle \sigma _{\mathrm {disc} }(S)}$ 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 ${\displaystyle X}$ be a Banach space and let ${\displaystyle T:\,D(T)\to X}$ be a closed linear operator on ${\displaystyle X}$ with dense domain ${\displaystyle D(T)}$. There are several definitions of the essential spectrum, which are not equivalent. ^[1]

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

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

${\displaystyle \sigma _{\mathrm {ess} ,1}(T)\subseteq \sigma _{\mathrm {ess} ,2}(T)\subseteq \sigma _{\mathrm {ess} ,3}(T)\subseteq \sigma _{\mathrm {ess} ,4}(T)\subseteq \sigma _{\mathrm {ess} ,5}(T)\subseteq \sigma (T)\subseteq \mathbb {C} ,}$

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

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

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

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

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

${\displaystyle \sigma _{\mathrm {ess} ,4}(T)=\bigcap _{K\in B_{0}(X)}\sigma (T+K),}$

where ${\displaystyle B_{0}(X)}$ denotes the set of compact operators on ${\displaystyle X}$ (D.E. Edmunds and W.D. Evans, 1987).

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

${\displaystyle \sigma (T)=\sigma _{\mathrm {ess} ,5}(T)\bigsqcup \sigma _{\mathrm {disc} }(T)}$,

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

See also

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

References

1. Gustafson, Karl (1969). "On the essential spectrum" (PDF). Journal of Mathematical Analysis and Applications. 25 (1): 121–127.

The self-adjoint case is discussed in

• Reed, Michael C.; Simon, Barry (1980), Methods of modern mathematical physics: Functional Analysis, vol. 1, San Diego: Academic Press, ISBN   0-12-585050-6
• Teschl, Gerald (2009). Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators. American Mathematical Society. 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 Annalen68, 220–269.