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".

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

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

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

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

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

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

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

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

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

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

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

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

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

has finite but non-zero dimension and that there is an ${\displaystyle \varepsilon >0}${\displaystyle \varepsilon >0} such that ${\displaystyle \mu \in \sigma (T)}${\displaystyle \mu \in \sigma (T)} and ${\displaystyle |\mu -\lambda |<\varepsilon }${\displaystyle |\mu -\lambda |<\varepsilon } imply that ${\displaystyle \mu }${\displaystyle \mu } and ${\displaystyle \lambda }${\displaystyle \lambda } are equal. (For general, non-self-adjoint operators ${\displaystyle S}${\displaystyle S} on Banach spaces, by definition, a complex number ${\displaystyle \lambda \in \mathbb {C} }${\displaystyle \lambda \in \mathbb {C} } is in the discrete spectrum ${\displaystyle \sigma _{\mathrm {disc} }(S)}${\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.)

Let ${\displaystyle X}${\displaystyle X} be a Banach space and let ${\displaystyle T:,円D(T)\to X}${\displaystyle T:,円D(T)\to X} be a closed linear operator on ${\displaystyle X}${\displaystyle X} with dense domain ${\displaystyle D(T)}${\displaystyle D(T)}. There are several definitions of the essential spectrum, which are not equivalent.^[1]

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

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

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

${\displaystyle r_{\mathrm {ess} ,k}(T)=\max\{|\lambda |:\lambda \in \sigma _{\mathrm {ess} ,k}(T)\}.}${\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 ${\displaystyle k=1,2,3,4,5}${\displaystyle k=1,2,3,4,5}.

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

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

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

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

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

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

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

• 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.

• Spectral theory

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