In mathematics, particularly in functional analysis, the spectrum of a bounded linear operator (or, more generally, an unbounded linear operator) is a generalisation of the set of eigenvalues of a matrix.
The study of spectra and related properties is known as spectral theory, which has numerous applications, most notably the mathematical formulation of quantum mechanics.
The spectrum of an operator on a finite-dimensional vector 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 the right shift operator R on the Hilbert space ℓ2, This has no eigenvalues, since if Rx=λx then by expanding this expression we see that x1=0, x2=0, etc.
On the other hand, 0 is in the spectrum because although the operator R − 0 (i.e. R itself) is invertible, the inverse is defined on a set which is not dense in ℓ2.
In fact every bounded linear operator on a complex Banach space must have a non-empty spectrum.
The notion of spectrum extends to unbounded (i.e. not necessarily bounded) operators.
, that consists of all bi-infinite sequences of real numbers that have a finite sum of squares
If the spectrum were empty, then the resolvent function would be defined everywhere on the complex plane and bounded.
By the vector-valued version of Liouville's theorem, this function is constant, thus everywhere zero as it is zero at infinity.
if the operator has a bounded everywhere-defined inverse, i.e. if there exists a bounded operator such that A complex number λ is then in the spectrum if λ is not in the resolvent set.
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.
Then, just as in the bounded case, a complex number λ lies in the spectrum of a closed operator T if and only if
The spectrum of an unbounded operator is in general a closed, possibly empty, subset of the complex plane.
The following subsections provide more details on the three parts of σ(T) sketched above.
The set of eigenvalues of T is also called the point spectrum of T, denoted by σp(T).
is an approximate eigenvalue; letting xn be the vector one can see that ||xn|| = 1 for all n, but Since R is a unitary operator, its spectrum lies on the unit circle.
By the spectral theorem, a bounded operator on a Hilbert space H is normal if and only if it is equivalent (after identification of H with an
The discrete spectrum is defined as the set of normal eigenvalues or, equivalently, as the set of isolated points of the spectrum such that the corresponding Riesz projector is of finite rank.
is injective and has dense range, but is not surjective, is called the continuous spectrum of T, denoted by
[6] There are five similar definitions of the essential spectrum of closed densely defined linear operator
since there are no eigenvalues embedded into the continuous spectrum) that can be computed by the Rydberg formula.
The result of the ionization process is described by the continuous part of the spectrum (the energy of the collision/ionization is not "quantized"), represented by
is the hermitian adjoint of T, then Theorem — For a bounded (or, more generally, closed and densely defined) operator T, In particular,
If T is a compact operator, or, more generally, an inessential operator, then it can be shown that the spectrum is countable, that zero is the only possible accumulation point, and that any nonzero λ in the spectrum is an eigenvalue.
For self-adjoint operators, one can use spectral measures to define a decomposition 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
fails to be invertible in the real algebra of bounded linear operators acting on
Let B be a complex Banach algebra containing a unit e. Then we define the spectrum σ(x) (or more explicitly σB(x)) of an element x of B to be the set of those complex numbers λ for which λe − x is not invertible in B.