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 $\lambda \in \mathbb {R}$ 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 $\lambda \in \mathbb {C}$ (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 $\sigma (T)$ . 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 $\{\psi _{k}\}_{k\in \mathbb {N} }\subseteq X$ 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 $\{\psi _{k}\}_{k\in \mathbb {N} }$ 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

\ 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 $\lambda \in \mathbb {C}$ 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.^[1]

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 $\mathbb {C} \setminus \sigma _{\mathrm {ess} ,1}(T)$ that do not intersect with the resolvent set $\mathbb {C} \setminus \sigma (T)$ .

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

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

^ 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 Annalen 68, 220–269.

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

[1]

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