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The essential spectrum of self-adjoint operators

In formal terms, let X {\displaystyle X} be a Hilbert space and let T {\displaystyle T} be a self-adjoint operator on X {\displaystyle X} .

Definition

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

T − λ I X {\displaystyle T-\lambda I_{X}}

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

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

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

lim k → ∞ ‖ ( T − λ ) ψ k ‖ = 0. {\displaystyle \lim _{k\to \infty }\left\|(T-\lambda )\psi _{k}\right\|=0.}

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

The discrete spectrum

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

σ d i s c ( T ) = σ ( T ) ∖ σ e s s ( T ) {\displaystyle \sigma _{\mathrm {disc} }(T)=\sigma (T)\setminus \sigma _{\mathrm {ess} }(T)} .

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

  s p a n { ψ ∈ X : T ψ = λ ψ } {\displaystyle \ \mathrm {span} \{\psi \in X:T\psi =\lambda \psi \}}

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

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

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

σ e s s , 1 ( T ) ⊆ σ e s s , 2 ( T ) ⊆ σ e s s , 3 ( T ) ⊆ σ e s s , 4 ( T ) ⊆ σ e s s , 5 ( T ) ⊆ σ ( T ) ⊆ C , {\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

r e s s , k ( T ) = max { | λ | : λ ∈ σ e s s , k ( T ) } . {\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 k = 1 , 2 , 3 , 4 , 5 {\displaystyle k=1,2,3,4,5} .

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

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

σ e s s , 4 ( T ) = ⋂ K ∈ B 0 ( X ) σ ( T + K ) , {\displaystyle \sigma _{\mathrm {ess} ,4}(T)=\bigcap _{K\in B_{0}(X)}\sigma (T+K),}

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

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

σ ( T ) = σ e s s , 5 ( T ) ⨆ σ d i s c ( T ) {\displaystyle \sigma (T)=\sigma _{\mathrm {ess} ,5}(T)\bigsqcup \sigma _{\mathrm {disc} }(T)} ,

where σ d i s c ( T ) {\displaystyle \sigma _{\mathrm {disc} }(T)} is the discrete spectrum of T {\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

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.

References

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