Inmathematics, particularly infunctional analysis, thespectrum of abounded linear operator (or, more generally, anunbounded linear operator) is a generalisation of the set ofeigenvalues of amatrix. Specifically, acomplex number${\displaystyle \lambda }$ is said to be in the spectrum of a bounded linear operator${\displaystyle T}$ if${\displaystyle T-\lambda I}$

• either hasno set-theoreticinverse;
• or the set-theoretic inverse is either unbounded or defined on a non-dense subset.^[1]

Here,${\displaystyle I}$ is theidentity operator.

By theclosed graph theorem,${\displaystyle \lambda }$ is in the spectrum if and only if the bounded operator${\displaystyle T-\lambda I:V\to V}$ is non-bijective on${\displaystyle V}$.

The study of spectra and related properties is known asspectral theory, which has numerous applications, most notably themathematical formulation of quantum mechanics.

The spectrum of an operator on afinite-dimensionalvector space is precisely the set of eigenvalues. However an operator on an infinite-dimensional space may have additional elements in its spectrum, and may have no eigenvalues. For example, consider theright shift operatorR on theHilbert spaceℓ²,

${\displaystyle (x_1,x_2,\dots )\mapsto (0,x_1,x_2,\dots ).}$

This has no eigenvalues, since ifRx=λx then by expanding this expression we see thatx₁=0,x₂=0, etc. On the other hand, 0 is in the spectrum because although the operatorR − 0 (i.e.R itself) is invertible, the inverse is defined on a set which is not dense inℓ². In factevery bounded linear operator on acomplexBanach space must have a non-empty spectrum.

The notion of spectrum extends tounbounded (i.e. not necessarily bounded) operators. Acomplex numberλ is said to be in the spectrum of an unbounded operator${\displaystyle T:X\to X}$ defined on domain${\displaystyle D(T)\subseteq X}$ if there is no bounded inverse${\displaystyle (T-\lambda I{)}^{-1}:X\to D(T)}$ defined on the whole of${\displaystyle X.}$ IfT isclosed (which includes the case whenT is bounded), boundedness of${\displaystyle (T-\lambda I{)}^{-1}}$ follows automatically from its existence.

The space of bounded linear operatorsB(X) on a Banach spaceX is an example of aunitalBanach algebra. Since the definition of the spectrum does not mention any properties ofB(X) except those that any such algebra has, the notion of a spectrum may be generalised to this context by using the same definition verbatim.

Let${\displaystyle T}$ be abounded linear operator acting on a Banach space${\displaystyle X}$ over the complex scalar field${\displaystyle \mathbb{C}}$, and${\displaystyle I}$ be theidentity operator on${\displaystyle X}$. Thespectrum of${\displaystyle T}$ is the set of all${\displaystyle \lambda \in \mathbb{C}}$ for which the operator${\displaystyle T-\lambda I}$ does not have an inverse that is a bounded linear operator.

Since${\displaystyle T-\lambda I}$ is a linear operator, the inverse is linear if it exists; and, by thebounded inverse theorem, it is bounded. Therefore, the spectrum consists precisely of those scalars${\displaystyle \lambda }$ for which${\displaystyle T-\lambda I}$ is notbijective.

The spectrum of a given operator${\displaystyle T}$ is often denoted${\displaystyle \sigma (T)}$, and its complement, theresolvent set, is denoted${\displaystyle \rho (T)=\mathbb{C}\setminus \sigma (T)}$. (${\displaystyle \rho (T)}$ is sometimes used to denote the spectral radius of${\displaystyle T}$)

If${\displaystyle \lambda }$ is an eigenvalue of${\displaystyle T}$, then the operator${\displaystyle T-\lambda I}$ is not one-to-one, and therefore its inverse${\displaystyle (T-\lambda I{)}^{-1}}$ is not defined. However, the converse statement is not true: the operator${\displaystyle T-\lambda I}$ may not have an inverse, even if${\displaystyle \lambda }$ is not an eigenvalue. Thus the spectrum of an operator always contains all its eigenvalues, but is not limited to them.

For example, consider the Hilbert space${\displaystyle {\ell }^2(\mathbb{Z})}$, that consists of allbi-infinite sequences of real numbers

${\displaystyle v=(\dots ,v_{-2},v_{-1},v_0,v_1,v_2,\dots )}$

that have a finite sum of squares$\sum _{i=-\mathrm{\infty }}^{+\mathrm{\infty }}{v}_i^2$. Thebilateral shift operator${\displaystyle T}$ simply displaces every element of the sequence by one position; namely if${\displaystyle u=T(v)}$ then${\displaystyle u_i=v_{i-1}}$ for every integer${\displaystyle i}$. The eigenvalue equation${\displaystyle T(v)=\lambda v}$ has no nonzero solution in this space, since it implies that all the values${\displaystyle v_i}$ have the same absolute value (if${\displaystyle |\lambda |=1}$) or are a geometric progression (if${\displaystyle |\lambda |\ne 1}$); either way, the sum of their squares would not be finite. However, the operator${\displaystyle T-\lambda I}$ is not invertible if${\displaystyle |\lambda |=1}$. For example, the sequence${\displaystyle u}$ such that${\displaystyle u_i=1/(|i|+1)}$ is in${\displaystyle {\ell }^2(\mathbb{Z})}$; but there is no sequence${\displaystyle v}$ in${\displaystyle {\ell }^2(\mathbb{Z})}$ such that${\displaystyle (T-I)v=u}$ (that is,${\displaystyle v_{i-1}=u_i+v_i}$ for all${\displaystyle i}$).

The spectrum of a bounded operator${\displaystyle T}$ is always aclosed,bounded subset of thecomplex plane.

If the spectrum were empty, then theresolvent function

${\displaystyle R(\lambda )=(T-\lambda I{)}^{-1},\lambda \in \mathbb{C},}$

would be defined everywhere on the complex plane and bounded. But it can be shown that the resolvent function${\displaystyle R}$ isholomorphic on its domain. By the vector-valued version ofLiouville's theorem, this function is constant, thus everywhere zero as it is zero at infinity. This would be a contradiction.

The boundedness of the spectrum follows from theNeumann series expansion in${\displaystyle \lambda }$; the spectrum${\displaystyle \sigma (T)}$ is bounded by${\displaystyle \Vert T\Vert }$. A similar result shows the closedness of the spectrum.

The bound${\displaystyle \Vert T\Vert }$ on the spectrum can be refined somewhat. Thespectral radius,${\displaystyle r(T)}$, of${\displaystyle T}$ is the radius of the smallest circle in the complex plane which is centered at the origin and contains the spectrum${\displaystyle \sigma (T)}$ inside of it, i.e.

${\displaystyle r(T)=sup\{|\lambda |:\lambda \in \sigma (T)\}.}$

Thespectral radius formula says^[2] that for any element${\displaystyle T}$ of aBanach algebra,

${\displaystyle r(T)=\underset{n\to \mathrm{\infty }}{lim}{\Vert T^n\Vert }^{1/n}.}$

One can extend the definition of spectrum tounbounded operators on aBanach spaceX. These operators are no longer elements in the Banach algebraB(X).

LetX be a Banach space and${\displaystyle T:D(T)\to X}$ be alinear operator defined on domain${\displaystyle D(T)\subseteq X}$.A complex numberλ is said to be in theresolvent set (also calledregular set) of${\displaystyle T}$ if the operator

${\displaystyle T-\lambda I:D(T)\to X}$

has a bounded everywhere-defined inverse, i.e. if there exists a bounded operator

${\displaystyle S:X\to D(T)}$

such that

${\displaystyle S(T-\lambda I)=I_{D(T)},(T-\lambda I)S=I_X.}$

A complex numberλ is then in thespectrum ifλ is not in the resolvent set.

Forλ to be in the resolvent (i.e. not in the spectrum), just like in the bounded case,${\displaystyle T-\lambda I}$ must be bijective, since it must have a two-sided inverse. As before, if an inverse exists, then its linearity is immediate, but in general it may not be bounded, so this condition must be checked separately.

By theclosed graph theorem, boundedness of${\displaystyle (T-\lambda I{)}^{-1}}$does follow directly from its existence whenT isclosed. Then, just as in the bounded case, a complex numberλ lies in the spectrum of a closed operatorT if and only if${\displaystyle T-\lambda I}$ is not bijective. Note that the class of closed operators includes all bounded operators.

The spectrum of an unbounded operator is in general a closed, possibly empty, subset of the complex plane.If the operatorT is notclosed, then${\displaystyle \sigma (T)=\mathbb{C}}$.

The following example indicates that non-closed operators may have empty spectra. Let${\displaystyle T}$ denote the differentiation operator on${\displaystyle L^2([0,1])}$, whose domain is defined to be the closure of${\displaystyle C_c^{\mathrm{\infty }}((0,1])}$ with respect to the${\displaystyle H^1}$-Sobolev space norm. This space can be characterized as all functions in${\displaystyle H^1([0,1])}$ that are zero at${\displaystyle t=0}$. Then,${\displaystyle T-z}$ has trivial kernel on this domain, as any${\displaystyle H^1([0,1])}$-function in its kernel is a constant multiple of${\displaystyle e^{zt}}$, which is zero at${\displaystyle t=0}$ if and only if it is identically zero. Therefore, the complement of the spectrum is all of${\displaystyle \mathbb{C}.}$

A bounded operatorT on a Banach space is invertible, i.e. has a bounded inverse, if and only ifT is bounded below, i.e.${\displaystyle \Vert Tx\Vert \ge c\Vert x\Vert ,}$ for some${\displaystyle c>0,}$ and has dense range. Accordingly, the spectrum ofT can be divided into the following parts:

1. ${\displaystyle \lambda \in \sigma (T)}$ if${\displaystyle T-\lambda I}$ is not bounded below. In particular, this is the case if${\displaystyle T-\lambda I}$ is not injective, that is,λ is an eigenvalue. The set of eigenvalues is called thepoint spectrum ofT and denoted byσ_p(T). Alternatively,${\displaystyle T-\lambda I}$ could be one-to-one but still not bounded below. Suchλ is not an eigenvalue but still anapproximate eigenvalue ofT (eigenvalues themselves are also approximate eigenvalues). The set of approximate eigenvalues (which includes the point spectrum) is called theapproximate point spectrum ofT, denoted byσ_ap(T).
2. ${\displaystyle \lambda \in \sigma (T)}$ if${\displaystyle T-\lambda I}$ does not have dense range. The set of suchλ is called thecompression spectrum ofT, denoted by${\displaystyle {\sigma }_{\mathrm{c}\mathrm{p}}(T)}$. If${\displaystyle T-\lambda I}$ does not have dense range but is injective,λ is said to be in theresidual spectrum ofT, denoted by${\displaystyle {\sigma }_{\mathrm{r}}(T)}$.

Note that the approximate point spectrum and residual spectrum are not necessarily disjoint^[3] (however, the point spectrum and the residual spectrum are).

The following subsections provide more details on the three parts ofσ(T) sketched above.

If an operator is not injective (so there is some nonzerox withT(x) = 0), then it is clearly not invertible. So ifλ is aneigenvalue ofT, one necessarily hasλ ∈ σ(T). The set of eigenvalues ofT is also called thepoint spectrum ofT, denoted byσ_p(T). Some authors refer to the closure of the point spectrum as thepure point spectrum${\displaystyle {\sigma }_{pp}(T)=\overline{{\sigma }_p(T)}}$ while others simply consider${\displaystyle {\sigma }_{pp}(T):={\sigma }_p(T).}$^[4][5]

More generally, by thebounded inverse theorem,T is not invertible if it is not bounded below; that is, if there is noc > 0 such that ||Tx|| ≥ c||x|| for allx ∈X. So the spectrum includes the set ofapproximate eigenvalues, which are thoseλ such thatT -λI is not bounded below; equivalently, it is the set ofλ for which there is a sequence of unit vectorsx₁,x₂, ... for which

${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\Vert T{x}_n-\lambda x_n\Vert =0}$.

The set of approximate eigenvalues is known as theapproximate point spectrum, denoted by${\displaystyle {\sigma }_{\mathrm{a}\mathrm{p}}(T)}$.

It is easy to see that the eigenvalues lie in the approximate point spectrum.

For example, consider the bilateral shiftW on${\displaystyle l^2(\mathbb{Z})}$ defined by

${\displaystyle W:e_j\mapsto e_{j+1},j\in \mathbb{Z},}$

where${\displaystyle (e_j{)}_{j\in \mathbb{N}}}$ is the standard orthonormal basis in${\displaystyle l^2(\mathbb{Z})}$. Direct calculation showsW has no eigenvalues, but everyλ with${\displaystyle |\lambda |=1}$ is an approximate eigenvalue; lettingx_n be the vector

${\displaystyle \frac{1}{\sqrt{n}}(\dots ,0,1,{\lambda }^{-1},{\lambda }^{-2},\dots ,{\lambda }^{1-n},0,\dots )}$

one can see that ||x_n|| = 1 for alln, but

${\displaystyle \Vert W{x}_n-\lambda x_n\Vert =\sqrt{\frac{2}{n}}\to 0.}$

SinceW is a unitary operator, its spectrum lies on the unit circle. Therefore, the approximate point spectrum ofW is its entire spectrum.

This conclusion is also true for a more general class of operators.A unitary operator isnormal. By thespectral theorem, a bounded operator on a Hilbert space H is normal if and only if it is equivalent (after identification ofH with an${\displaystyle L^2}$ space) to amultiplication operator. It can be shown that the approximate point spectrum of a bounded multiplication operator equals its spectrum.

Thediscrete spectrum is defined as the set ofnormal eigenvalues or, equivalently, as the set of isolated points of the spectrum such that the correspondingRiesz projector is of finite rank. As such, the discrete spectrum is a strict subset of the point spectrum, i.e.,${\displaystyle {\sigma }_d(T)\subset {\sigma }_p(T).}$

The set of allλ for which${\displaystyle T-\lambda I}$ is injective and has dense range, but is not surjective, is called thecontinuous spectrum ofT, denoted by${\displaystyle {\sigma }_{\mathbb{c}}(T)}$. The continuous spectrum therefore consists of those approximate eigenvalues which are not eigenvalues and do not lie in the residual spectrum. That is,

${\displaystyle {\sigma }_{\mathrm{c}}(T)={\sigma }_{\mathrm{a}\mathrm{p}}(T)\setminus ({\sigma }_{\mathrm{r}}(T)\cup {\sigma }_{\mathrm{p}}(T))}$.

For example,${\displaystyle A:l^2(\mathbb{N})\to l^2(\mathbb{N})}$,${\displaystyle e_j\mapsto e_j/j}$,${\displaystyle j\in \mathbb{N}}$, is injective and has a dense range, yet${\displaystyle \mathrm{R}\mathrm{a}\mathrm{n}(A)⊊l^2(\mathbb{N})}$.Indeed, if$x=\sum _{j\in \mathbb{N}}{c}_j{e}_j\in l^2(\mathbb{N})$ with${\displaystyle c_j\in \mathbb{C}}$ such that, one does not necessarily have, and then$\sum _{j\in \mathbb{N}}j{c}_j{e}_j\notin l^2(\mathbb{N})$.

The set of${\displaystyle \lambda \in \mathbb{C}}$ for which${\displaystyle T-\lambda I}$ does not have dense range is known as thecompression spectrum ofT and is denoted by${\displaystyle {\sigma }_{\mathrm{c}\mathrm{p}}(T)}$.

The set of${\displaystyle \lambda \in \mathbb{C}}$ for which${\displaystyle T-\lambda I}$ is injective but does not have dense range is known as theresidual spectrum ofT and is denoted by${\displaystyle {\sigma }_{\mathrm{r}}(T)}$:

${\displaystyle {\sigma }_{\mathrm{r}}(T)={\sigma }_{\mathrm{c}\mathrm{p}}(T)\setminus {\sigma }_{\mathrm{p}}(T).}$

An operator may be injective, even bounded below, but still not invertible. The right shift on${\displaystyle l^2(\mathbb{N})}$,${\displaystyle R:l^2(\mathbb{N})\to l^2(\mathbb{N})}$,${\displaystyle R:e_j\mapsto e_{j+1},j\in \mathbb{N}}$, is such an example. This shift operator is anisometry, therefore bounded below by 1. But it is not invertible as it is not surjective (${\displaystyle e_1\notin \mathrm{R}\mathrm{a}\mathrm{n}(R)}$), and moreover${\displaystyle \mathrm{R}\mathrm{a}\mathrm{n}(R)}$ is not dense in${\displaystyle l^2(\mathbb{N})}$(${\displaystyle e_1\notin \overline{\mathrm{R}\mathrm{a}\mathrm{n}(R)}}$).

The peripheral spectrum of an operator is defined as the set of points in its spectrum which have modulus equal to its spectral radius.^[6]

There are five similar definitions of theessential spectrum of closed densely defined linear operator${\displaystyle A:X\to X}$ which satisfy

${\displaystyle {\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s},1}(A)\subset {\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s},2}(A)\subset {\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s},3}(A)\subset {\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s},4}(A)\subset {\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s},5}(A)\subset \sigma (A).}$

All these spectra${\displaystyle {\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s},k}(A),1\le k\le 5}$, coincide in the case of self-adjoint operators.

1. The essential spectrum${\displaystyle {\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s},1}(A)}$ is defined as the set of points${\displaystyle \lambda }$ of the spectrum such that${\displaystyle A-\lambda I}$ is notsemi-Fredholm. (The operator issemi-Fredholm if its range is closed and either its kernel or cokernel (or both) is finite-dimensional.)
   Example 1:${\displaystyle \lambda =0\in {\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s},1}(A)}$ for the operator${\displaystyle A:l^2(\mathbb{N})\to l^2(\mathbb{N})}$,${\displaystyle A:e_j\mapsto e_j/j,j\in \mathbb{N}}$ (because the range of this operator is not closed: the range does not include all of${\displaystyle l^2(\mathbb{N})}$ although its closure does).
   Example 2:${\displaystyle \lambda =0\in {\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s},1}(N)}$ for${\displaystyle N:l^2(\mathbb{N})\to l^2(\mathbb{N})}$,${\displaystyle N:v\mapsto 0}$ for any${\displaystyle v\in l^2(\mathbb{N})}$ (because both kernel and cokernel of this operator are infinite-dimensional).
2. The essential spectrum${\displaystyle {\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s},2}(A)}$ is defined as the set of points${\displaystyle \lambda }$ of the spectrum such that the operator either${\displaystyle A-\lambda I}$ has infinite-dimensional kernel or has a range which is not closed. It can also be characterized in terms ofWeyl's criterion: there exists asequence${\displaystyle (x_j{)}_{j\in \mathbb{N}}}$ in the spaceX such that${\displaystyle \Vert x_j\Vert =1}$,$\underset{j\to \mathrm{\infty }}{lim}\Vert (A-\lambda I)x_j\Vert =0,$ and such that${\displaystyle (x_j{)}_{j\in \mathbb{N}}}$ contains no convergentsubsequence. Such a sequence is called asingular sequence (or asingular Weyl sequence).
   Example:${\displaystyle \lambda =0\in {\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s},2}(B)}$ for the operator${\displaystyle B:l^2(\mathbb{N})\to l^2(\mathbb{N})}$,${\displaystyle B:e_j\mapsto e_{j/2}}$ ifj is even and${\displaystyle e_j\mapsto 0}$ whenj is odd (kernel is infinite-dimensional; cokernel is zero-dimensional). Note that${\displaystyle \lambda =0\notin {\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s},1}(B)}$.
3. The essential spectrum${\displaystyle {\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s},3}(A)}$ is defined as the set of points${\displaystyle \lambda }$ of the spectrum such that${\displaystyle A-\lambda I}$ is notFredholm. (The operator isFredholm if its range is closed and both its kernel and cokernel are finite-dimensional.)
   Example:${\displaystyle \lambda =0\in {\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s},3}(J)}$ for the operator${\displaystyle J:l^2(\mathbb{N})\to l^2(\mathbb{N})}$,${\displaystyle J:e_j\mapsto e_{2j}}$ (kernel is zero-dimensional, cokernel is infinite-dimensional). Note that${\displaystyle \lambda =0\notin {\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s},2}(J)}$.
4. The essential spectrum${\displaystyle {\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s},4}(A)}$ is defined as the set of points${\displaystyle \lambda }$ of the spectrum such that${\displaystyle A-\lambda I}$ is notFredholm of index zero. It could also be characterized as the largest part of the spectrum ofA which is preserved bycompact perturbations. In other words,${\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s},4}(A)=\bigcap _{K\in B_0(X)}\sigma (A+K)$; here${\displaystyle B_0(X)}$ denotes the set of all compact operators onX.
   Example:${\displaystyle \lambda =0\in {\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s},4}(R)}$ where${\displaystyle R:l^2(\mathbb{N})\to l^2(\mathbb{N})}$ is the right shift operator,${\displaystyle R:l^2(\mathbb{N})\to l^2(\mathbb{N})}$,${\displaystyle R:e_j\mapsto e_{j+1}}$ for${\displaystyle j\in \mathbb{N}}$ (its kernel is zero, its cokernel is one-dimensional). Note that${\displaystyle \lambda =0\notin {\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s},3}(R)}$.
5. The essential spectrum${\displaystyle {\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s},5}(A)}$ is the union of${\displaystyle {\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s},1}(A)}$ with all components of${\displaystyle \mathbb{C}\setminus {\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s},1}(A)}$ that do not intersect with the resolvent set${\displaystyle \mathbb{C}\setminus \sigma (A)}$. It can also be characterized as${\displaystyle \sigma (A)\setminus {\sigma }_{\mathrm{d}}(A)}$.
   Example: consider the operator${\displaystyle T:l^2(\mathbb{Z})\to l^2(\mathbb{Z})}$,${\displaystyle T:e_j\mapsto e_{j-1}}$ for${\displaystyle j\ne 0}$,${\displaystyle T:e_0\mapsto 0}$. Since${\displaystyle \Vert T\Vert =1}$, one has${\displaystyle \sigma (T)\subset \overline{{\mathbb{D}}_1}}$. For any${\displaystyle z\in \mathbb{C}}$ with${\displaystyle |z|=1}$, the range of${\displaystyle T-zI}$ is dense but not closed, hence the boundary of the unit disc is in the first type of the essential spectrum:${\displaystyle \mathrm{\partial }{\mathbb{D}}_1\subset {\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s},1}(T)}$. For any${\displaystyle z\in \mathbb{C}}$ with,${\displaystyle T-zI}$ has a closed range, one-dimensional kernel, and one-dimensional cokernel, so${\displaystyle z\in \sigma (T)}$ although${\displaystyle z\notin {\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s},k}(T)}$ for${\displaystyle 1\le k\le 4}$; thus,${\displaystyle {\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s},k}(T)=\mathrm{\partial }{\mathbb{D}}_1}$ for${\displaystyle 1\le k\le 4}$. There are two components of${\displaystyle \mathbb{C}\setminus {\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s},1}(T)}$:${\displaystyle \{z\in \mathbb{C}:|z|>1\}}$ and. The component has no intersection with the resolvent set; by definition,.

Thehydrogen atom provides an example of different types of the spectra. Thehydrogen atom Hamiltonian operator${\displaystyle H=-\mathrm{\Delta }-\frac{Z}{|x|}}$,${\displaystyle Z>0}$, with domain${\displaystyle D(H)=H^1({\mathbb{R}}^3)}$ has a discrete set of eigenvalues (the discrete spectrum${\displaystyle {\sigma }_{\mathrm{d}}(H)}$, which in this case coincides with the point spectrum${\displaystyle {\sigma }_{\mathrm{p}}(H)}$ since there are no eigenvalues embedded into the continuous spectrum) that can be computed by theRydberg formula. Their correspondingeigenfunctions are calledeigenstates, or thebound states. The result of theionization process is described by the continuous part of the spectrum (the energy of the collision/ionization is not "quantized"), represented by${\displaystyle {\sigma }_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}}(H)=[0,+\mathrm{\infty })}$ (it also coincides with the essential spectrum,${\displaystyle {\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s}}(H)=[0,+\mathrm{\infty })}$).^{[citation needed][clarification needed]}

LetX be a Banach space and${\displaystyle T:X\to X}$ aclosed linear operator with dense domain${\displaystyle D(T)\subset X}$.IfX* is the dual space ofX, and${\displaystyle T^{\ast }:X^{\ast }\to X^{\ast }}$ is thehermitian adjoint ofT, then

${\displaystyle \sigma (T^{\ast })=\overline{\sigma (T)}:=\{z\in \mathbb{C}:\overline{z}\in \sigma (T)\}.}$

We also get${\displaystyle {\sigma }_{\mathrm{p}}(T)\subset \overline{{\sigma }_{\mathrm{r}}(T^{\ast })\cup {\sigma }_{\mathrm{p}}(T^{\ast })}}$ by the following argument:X embeds isometrically intoX**. Therefore, for every non-zero element in the kernel of${\displaystyle T-\lambda I}$ there exists a non-zero element inX** which vanishes on${\displaystyle \mathrm{R}\mathrm{a}\mathrm{n}(T^{\ast }-\overline{\lambda }I)}$. Thus${\displaystyle \mathrm{R}\mathrm{a}\mathrm{n}(T^{\ast }-\overline{\lambda }I)}$ can not be dense.

Furthermore, ifX is reflexive, we have${\displaystyle \overline{{\sigma }_{\mathrm{r}}(T^{\ast })}\subset {\sigma }_{\mathrm{p}}(T)}$.

IfT is acompact operator, or, more generally, aninessential operator, then it can be shown that the spectrum is countable, that zero is the only possibleaccumulation point, and that any nonzeroλ in the spectrum is an eigenvalue.

A bounded operator${\displaystyle A:X\to X}$ isquasinilpotent if${\displaystyle \Vert A^n{\Vert }^{1/n}\to 0}$ as${\displaystyle n\to \mathrm{\infty }}$ (in other words, if the spectral radius ofA equals zero). Such operators could equivalently be characterized by the condition

${\displaystyle \sigma (A)=\{0\}.}$

An example of such an operator is${\displaystyle A:l^2(\mathbb{N})\to l^2(\mathbb{N})}$,${\displaystyle e_j\mapsto e_{j+1}/2^j}$ for${\displaystyle j\in \mathbb{N}}$.

IfX is aHilbert space andT is aself-adjoint operator (or, more generally, anormal operator), then a remarkable result known as thespectral theorem gives an analogue of the diagonalisation theorem for normal finite-dimensional operators (Hermitian matrices, for example).

For self-adjoint operators, one can usespectral measures to define adecomposition of the spectrum into absolutely continuous, pure point, and singular parts.

The definitions of the resolvent and spectrum can be extended to any continuous linear operator${\displaystyle T}$ acting on a Banach space${\displaystyle X}$ over the real field${\displaystyle \mathbb{R}}$ (instead of the complex field${\displaystyle \mathbb{C}}$) via itscomplexification${\displaystyle T_{\mathbb{C}}}$. In this case we define the resolvent set${\displaystyle \rho (T)}$ as the set of all${\displaystyle \lambda \in \mathbb{C}}$ such that${\displaystyle T_{\mathbb{C}}-\lambda I}$ is invertible as an operator acting on the complexified space${\displaystyle X_{\mathbb{C}}}$; then we define${\displaystyle \sigma (T)=\mathbb{C}\setminus \rho (T)}$.

Thereal spectrum of a continuous linear operator${\displaystyle T}$ acting on a real Banach space${\displaystyle X}$, denoted${\displaystyle {\sigma }_{\mathbb{R}}(T)}$, is defined as the set of all${\displaystyle \lambda \in \mathbb{R}}$ for which${\displaystyle T-\lambda I}$ fails to be invertible in the real algebra of bounded linear operators acting on${\displaystyle X}$. In this case we have${\displaystyle \sigma (T)\cap \mathbb{R}={\sigma }_{\mathbb{R}}(T)}$. Note that the real spectrum may or may not coincide with the complex spectrum. In particular, the real spectrum could be empty.

This sectionneeds expansion. You can help byadding missing information.(June 2009)

LetB be a complexBanach algebra containing aunite. Then we define the spectrumσ(x) (or more explicitlyσ_B(x)) of an elementx ofB to be the set of thosecomplex numbersλ for whichλe − x is not invertible inB. This extends the definition for bounded linear operatorsB(X) on a Banach spaceX, sinceB(X) is a unital Banach algebra.

• Essential spectrum
• Discrete spectrum (mathematics)
• Self-adjoint operator
• Pseudospectrum
• Resolvent set

• Dales et al.,Introduction to Banach Algebras, Operators, and Harmonic Analysis,ISBN 0-521-53584-0
• "Spectrum of an operator",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
• Simon, Barry (2005).Orthogonal polynomials on the unit circle. Part 1. Classical theory. American Mathematical Society Colloquium Publications. Vol. 54. Providence, R.I.:American Mathematical Society.ISBN 978-0-8218-3446-6.MR 2105088.
• Teschl, G. (2014).Mathematical Methods in Quantum Mechanics. Providence (R.I): American Mathematical Soc.ISBN 978-1-4704-1704-8.

• Spectral theory
• Functional analysis

• Articles with short description
• Short description is different from Wikidata
• All articles with unsourced statements
• Articles with unsourced statements from August 2019
• Wikipedia articles needing clarification from November 2023
• Articles to be expanded from June 2009
• All articles to be expanded
• Pages that use a deprecated format of the math tags