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".

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 or Weyl 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.)

Examples

Let ${\displaystyle T:L^{2}[0,1]\to L^{2}[0,1]}$ be the multiplication operator defined by ${\displaystyle (Tf)(x)=xf(x)}$. The essential range of ${\displaystyle x\mapsto x}$ is ${\displaystyle [0,1]}$, so the spectrum is ${\displaystyle \sigma (T)=[0,1]}$. For any ${\displaystyle \lambda \in [0,1]}$, we can explicitly construct a singular sequence as a sequence of increasingly narrow and sharp rectangular functions that are supported on disjoint sets. For example, let ${\displaystyle \lambda =0}$, then we can construct ${\displaystyle \psi _{n}}$ to be the rectangular function on ${\displaystyle [2^{-n},2^{-n+1}]}$ of height ${\displaystyle {\sqrt {2^{n}}}}$.

Of densely defined operators

Preliminary concepts

Let ${\displaystyle X}$ be a Banach space, and let ${\displaystyle T}$ be a densely defined operator on ${\displaystyle X}$. That is, it is of type ${\displaystyle T:\,D(T)\to X}$, where ${\displaystyle D(T)}$ is a dense subspace of ${\displaystyle X}$. Let the spectrum of ${\displaystyle T}$ be ${\displaystyle \sigma (T)}$, defined by$${\displaystyle \sigma (T)=\{\lambda$$ :(\lambda I-T){\text{ has no bounded inverse}}\}} The complement of ${\displaystyle \sigma (T)}$ is the resolvent set of ${\displaystyle T}$.

Definitions

There are several definitions of the essential spectrum of ${\displaystyle T}$, which are not necessarily the same. Each of these definitions is of the form$${\displaystyle \sigma _{\mathrm {ess} }(T)=\{\lambda$$ :(\lambda I-T){\text{ is not nice}}\}} There are at least 5 different levels of niceness, increasing in strength. Each increase in strength shrinks the set of nice ${\displaystyle \lambda }$, thus expands the essential domain.^[1]

Let ${\displaystyle A}$ denote an operator of type ${\displaystyle A:D(T)\to X}$. Let ${\displaystyle \ker A}$ be its kernel, ${\displaystyle \operatorname {coker} A}$ be its cokernel, ${\displaystyle \operatorname {ran} A}$ be its range. We say that ${\displaystyle A}$ is:

1. Normally solvable, if ${\displaystyle A}$ is a closed operator, and ${\displaystyle \operatorname {ran} A}$ is a closed set. This can be checked via the closed range theorem.
2. Semi-Fredholm, if furthermore, ${\displaystyle \ker A}$ is finite-dimensional inclusive-or ${\displaystyle \operatorname {coker} A}$ is finite-dimensional.
3. Fredholm, if furthermore, ${\displaystyle \ker A}$ is finite-dimensional and ${\displaystyle \operatorname {coker} A}$ is finite-dimensional.
4. Fredholm with index zero, if furthermore, ${\displaystyle \ker A}$ and ${\displaystyle \operatorname {coker} A}$ has the same dimension.
5. If furthermore, there exists a deleted neighborhood of zero that is a subset of the resolvent set.
   • In other words, zero is not a limit point of ${\displaystyle \sigma (A)}$.
6. Has bounded inverse, if there exists a bounded linear operator ${\displaystyle A^{-1}:X\to D(T)}$, such that ${\displaystyle A,A^{-1}}$ are inverses of each other.

Now, set ${\displaystyle A=(\lambda I-T)}$. Then conditions 1 to 5 defines 5 essential spectra ${\displaystyle \sigma _{\mathrm {ess} ,k}(T)}$, ${\displaystyle 1\leq k\leq 5}$, and condition 6 defines the spectrum ${\displaystyle \sigma (T)}$. It is clear that conditions 1 to 5 increases in strength. One can also show that condition 6 is stronger than condition 5. Thus,$${\displaystyle \sigma _{\mathrm {ess} ,1}(T)\subseteq \sigma _{\mathrm {ess} ,2}(T)\subseteq \sigma _{\mathrm {ess} ,3}(T)\subseteq \sigma _{\mathrm {ess} ,4}(T)\subseteq \sigma _{\mathrm {ess} ,5}(T)\subseteq \sigma (T)\subseteq \mathbb {C} ,}$$Any of these inclusions may be strict.

Different authors defined the essential spectra differently, resulting in different terminologies. For example, Kato used ${\displaystyle \sigma _{\mathrm {ess} ,2}}$, Wolf used ${\displaystyle \sigma _{\mathrm {ess} ,3}}$, Schechter used ${\displaystyle \sigma _{\mathrm {ess} ,4}}$, Browder used ${\displaystyle \sigma _{\mathrm {ess} ,5}}$. Thus, ${\displaystyle \sigma _{\mathrm {ess} ,5}}$ is also called the Browder essential spectrum, etc.^[2]

More definitions

There are even more definitions of the essential spectrum.^[1]

The following definition states that the essential spectrum is the part of the spectrum that is stable under compact perturbation:$${\displaystyle w(T):=\bigcap _{B{\text{ is compact}}}\sigma (T+B)}$$Another definition states that:$${\displaystyle \sigma _{l}(T)=\sigma (T)\setminus \{{\text{isolated eigenvalues of }}T{\text{ with finite multiplicity}}\}}$$Given ${\displaystyle \lambda \in \mathbb {C} }$, it is an isolated eigenvalue of ${\displaystyle T}$ with finite multiplicity if and only if ${\displaystyle \ker(\lambda I-T)}$ has positive finite dimension, and ${\displaystyle \lambda }$ is an isolated point of ${\displaystyle \sigma (T)}$.

Equalities

Banach space case

1. If ${\displaystyle T}$ is not closed, then ${\displaystyle \sigma _{\mathrm {ess} ,1}(T)=\mathbb {C} }$. Because of this, the essential spectrum is uninteresting for these, and we will assume thenceforth that ${\displaystyle T}$ is closed.
2. If ${\displaystyle T}$ is bounded and either hypernormal or Toeplitz, then ${\displaystyle \sigma _{\mathrm {ess} ,4}(T)=\sigma _{\mathrm {ess} ,5}(T)}$.
3. If ${\displaystyle T}$ is bounded and ${\displaystyle \sigma _{\mathrm {ess} ,2}(T)}$, then ${\displaystyle \sigma _{\mathrm {ess} ,2}(T)=\sigma _{\mathrm {ess} ,5}(T)}$.
4. ${\displaystyle \sigma _{\mathrm {ess} ,k}(T')=\sigma _{\mathrm {ess} ,k}(T)}$ for all ${\displaystyle k=1,2,3,4,5}$, where ${\displaystyle T'}$ is the transpose operator of ${\displaystyle T}$.
5. 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 ${\displaystyle k=1,2,3,4,5}$.
6. The essential spectrum ${\displaystyle \sigma _{\mathrm {ess} ,k}(T)}$ is invariant under compact perturbations for ${\displaystyle k=1,2,3,4}$, but not for ${\displaystyle k=5}$. That is, for ${\displaystyle k=1,2,3,4}$ and any compact operator ${\displaystyle B}$, ${\displaystyle \sigma _{\mathrm {ess} ,k}(T+B)=\sigma _{\mathrm {ess} ,k}(T)}$. The 4th essential spectrum is in fact the maximal possible that is stable under compact perturbations, in the sense that ${\displaystyle \sigma _{\mathrm {ess} ,4}(T)=w(T)}$. (D.E. Edmunds and W.D. Evans, 1987).
7. ${\displaystyle \sigma _{\mathrm {ess} ,5}(T)=\sigma _{l}(T)}$.
8. ${\displaystyle \sigma (T)=\sigma _{\mathrm {ess} ,5}(T)\bigsqcup \sigma _{\mathrm {disc} }(T)}$, where ${\displaystyle \sigma _{\mathrm {disc} }(T)}$ is the discrete spectrum of ${\displaystyle T}$.

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.

Hilbert space case

If ${\displaystyle X}$ is a Hilbert space, and ${\displaystyle T}$ is self-adjoint, then all the above definitions of the essential spectrum coincide, except ${\displaystyle \sigma _{\mathrm {ess} ,1}(T)}$. Concretely, we have^[1]$${\displaystyle \sigma _{\mathrm {ess} ,1}(T)\subseteq \sigma _{\mathrm {ess} ,2}(T)=\sigma _{\mathrm {ess} ,3}(T)=\sigma _{\mathrm {ess} ,4}(T)=\sigma _{\mathrm {ess} ,5}(T)}$$The issue is that ${\displaystyle \sigma _{\mathrm {ess} ,1}(T)}$ does not include isolated eigenvalues of infinite multiplicity. For example, if ${\displaystyle T=I}$ and ${\displaystyle X}$ is infinite-dimensional, then ${\displaystyle \sigma _{\mathrm {ess} ,1}(T)}$ is empty, whereas ${\displaystyle \sigma (T)=\{1\}}$. This is because 1 is an eigenvalue of the identity operator with infinite multiplicity.

If ${\displaystyle X}$ is a Hilbert space, then ${\displaystyle \sigma _{\mathrm {ess} ,k}(T^{*})={\overline {\sigma _{\mathrm {ess} ,k}(T)}}}$ for all ${\displaystyle k=1,2,3,4,5}$.

See also

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