Wikipedia / Spectrum (functional analysis)
Wikipedia / Spectrum (functional analysis)
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. Specifically, a complex number
either has no set-theoretic inverse;
or the set-theoretic inverse is either unbounded or defined on a non-dense subset.
is the identity operator.
By the closed graph theorem.
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. A complex number λ If T is closed (which includes the case when T follows automatically from its existence.
The space of bounded linear operators B(X) on a Banach space X is an example of a unital Banach algebra. Since the definition of the spectrum does not mention any properties of B(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.
Spectrum of a bounded operator
Definition
be a bounded linear operator be the identity operator. The spectrum does not have an inverse that is a bounded linear operator.
is a linear operator, the inverse is linear if it exists; and, by the bounded inverse theorem is not bijective.
, and its complement, the resolvent set)
Relation to eigenvalues
is not an eigenvalue. Thus the spectrum of an operator always contains all its eigenvalues, but is not limited to them.
, that consists of all bi-infinite sequences of real numbers
. The bilateral shift).
Basic properties
is always a closed, bounded subset of the complex plane.
If the spectrum were empty, then the resolvent function
is holomorphic on its domain. By the vector-valued version of Liouville'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 the Neumann series expansion. A similar result shows the closedness of the spectrum.
on the spectrum can be refined somewhat. The spectral radius inside of it, i.e.
The spectral radius formula says of a Banach algebra,
Spectrum of an unbounded operator
One can extend the definition of spectrum to unbounded operators on a Banach space X. These operators are no longer elements in the Banach algebra B(X).
Definition
Let X be a linear operator. A complex number λ is said to be in the resolvent set (also called regular set 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.
For λ 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 the closed graph theorem does follow directly from its existence when T is closed. Then, just as in the bounded case, a complex number λ lies in the spectrum of a closed operator T is not bijective. Note that the class of closed operators includes all bounded operators.
Basic properties
The spectrum of an unbounded operator is in general a closed, possibly empty, subset of the complex plane. If the operator T is not closed.
Classification of points in the spectrum
A bounded operator T on a Banach space is invertible, i.e. has a bounded inverse, if and only if T and has dense range. Accordingly, the spectrum of T can be divided into the following parts:
is not injective, that is, λ is an eigenvalue. The set of eigenvalues is called the point spectrum of T and denoted by σp(T could be one-to-one but still not bounded below. Such λ is not an eigenvalue but still an approximate eigenvalue of T (eigenvalues themselves are also approximate eigenvalues). The set of approximate eigenvalues (which includes the point spectrum) is called the approximate point spectrum of T, denoted by σap(T).
does not have dense range. The set of such λ is called the compression spectrum of T does not have dense range but is injective, λ is said to be in the residual spectrum of T.
Note that the approximate point spectrum and residual spectrum are not necessarily disjoint (however, the point spectrum and the residual spectrum are).
The following subsections provide more details on the three parts of σ(T) sketched above.
Point spectrum
If an operator is not injective (so there is some nonzero x with T(x)=0), then it is clearly not invertible. So if λ is an eigenvalue of T, one necessarily has λ∈σ(T). The set of eigenvalues of T is also called the point spectrum of T, denoted by σp(T). Some authors refer to the closure of the point spectrum as the pure point spectrum
Approximate point spectrum
More generally, by the bounded inverse theorem, T is not invertible if it is not bounded below; that is, if there is no c>0 such that ||Tx||≥c||x|| for all . So the spectrum includes the set of approximate eigenvalues, which are those λ such that is not bounded below; equivalently, it is the set of λ for which there is a sequence of unit vectors x1, x2, ... for which
.
The set of approximate eigenvalues is known as the approximate point spectrum.
It is easy to see that the eigenvalues lie in the approximate point spectrum.
For example, consider the bilateral shift W defined by
. Direct calculation shows W has no eigenvalues, but every λ is an approximate eigenvalue; letting xn be the vector
one can see that ||xn|| = 1 for all n, but
Since W is a unitary operator, its spectrum lies on the unit circle. Therefore, the approximate point spectrum of W is its entire spectrum.
This conclusion is also true for a more general class of operators. A unitary operator is normal. 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 space) to a multiplication operator. It can be shown that the approximate point spectrum of a bounded multiplication operator equals its spectrum.
Discrete spectrum
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
Continuous spectrum
The set of all λ is injective and has dense range, but is not surjective, is called the continuous spectrum of T. The continuous spectrum therefore consists of those approximate eigenvalues which are not eigenvalues and do not lie in the residual spectrum. That is,
.
.
Compression spectrum
does not have dense range is known as the compression spectrum of T.
Residual spectrum
is injective but does not have dense range is known as the residual spectrum of T:
, is such an example. This shift operator is an isometry).
Peripheral spectrum
The peripheral spectrum of an operator is defined as the set of points in its spectrum which have modulus equal to its spectral radius.
Essential spectrum
There are five similar definitions of the essential spectrum which satisfy
, coincide in the case of self-adjoint operators.
is not semi-Fredholm. (The operator is semi-Fredholm if its range is closed and either its kernel or cokernel (or both) is finite-dimensional.) Example 1: although its closure does).Example 2: (because both kernel and cokernel of this operator are infinite-dimensional).
has infinite-dimensional kernel or has a range which is not closed. It can also be characterized in terms of Weyl's criterion: there exists a sequence in the space X contains no convergent subsequence. Such a sequence is called a singular sequence (or a singular Weyl sequence).Example: if j when j.
is not Fredholm. (The operator is Fredholm if its range is closed and both its kernel and cokernel are finite-dimensional.) Example:.
is not Fredholm of index zero. It could also be characterized as the largest part of the spectrum of A which is preserved by compact denotes the set of all compact operators on X. Example:.
.Example:.
Example: Hydrogen atom
The hydrogen atom provides an example of different types of the spectra. The hydrogen atom Hamiltonian operator since there are no eigenvalues embedded into the continuous spectrum) that can be computed by the Rydberg formula. Their corresponding eigenfunctions are called eigenstates, or the bound states. The result of the ionization).
Spectrum of the adjoint operator
Let X a closed linear operator. If X* is the dual space of X is the hermitian adjoint of T, then
by the following argument: X embeds isometrically into X** there exists a non-zero element in X** can not be dense.
Furthermore, if X.
Spectra of particular classes of operators
Compact operators
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.
Quasinilpotent operators
is quasinilpotent (in other words, if the spectral radius of A equals zero). Such operators could equivalently be characterized by the condition
.
Self-adjoint operators
If X is a Hilbert space and T is a self-adjoint operator (or, more generally, a normal operator), then a remarkable result known as the spectral theorem gives an analogue of the diagonalisation theorem for normal finite-dimensional operators (Hermitian matrices, for example).
For self-adjoint operators, one can use spectral measures to define a decomposition of the spectrum into absolutely continuous, pure point, and singular parts.
Spectrum of a real operator
) via its complexification.
Real spectrum
The real spectrum. Note that the real spectrum may or may not coincide with the complex spectrum. In particular, the real spectrum could be empty.
Spectrum of a unital Banach algebra
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. This extends the definition for bounded linear operators B(X) on a Banach space X, since B(X) is a unital Banach algebra.
See also
Notes
References
Dales et al., Introduction to Banach Algebras, Operators, and Harmonic Analysis,