Spectrum (functional analysis)
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In functional analysis, the concept of the spectrum of an element of a Banach algebra is a generalisation of the concept of eigenvalues, which is exceedingly useful in the case of operators on infinite-dimensional spaces. For example, the bilateral shift operator on the Hilbert space <math>\ell^2(\mathbf Z)</math> has no eigenvalues at all; but we shall see below that any bounded linear operator on a complex Banach space must have non-empty spectrum.
The study of the properties of spectra is known as spectral theory.
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Definition
Let B be a complex Banach algebra containing a unit e. The spectrum of an element x of B, often written as σB(x) or simply σ(x), consists of those complex numbers λ for which λ e - x is not invertible in B.
Basic properties
The spectrum σ(x) of an element x of B is always compact and non-empty. If the spectrum were empty, then the resolvent function
- R(λ) = (λe - x)-1
would be defined everywhere on the complex plane and bounded, which would imply by Liouville's theorem that this function is constant, thus everywhere zero as it is zero at infinity, which would be a contradiction. The boundedness of the spectrum follows from the Neumann series expansion in λ, which also helps prove the closedness of the spectrum and hence its compactness.
One defines the spectral radius to be the radius of the smallest circle in the complex plane which is centered at the origin and contains the spectrum σ(x) inside of it. This is equivalent to the supremum of the absolute values of the members of the spectrum:
- <math>r(x) = \sup \{|\lambda| : \lambda \in \sigma(x)\}.</math>
One can prove that r(x) is bounded above by ||x||. The spectral radius formula is a refinement of this statement. It says that
- <math>r(x) = \lim_{n \to \infty} \|x^n\|^{1/n}.</math>
Spectrum of a bounded operator
If X is a complex Banach space, then the set of all bounded linear operators on X forms a Banach algebra, denoted by B(X). This makes it possible to define the spectrum of a bounded linear operator viewed as an element in this Banach algebra. More specifically, denote by I the identity operator on X, so that I is the unit of B(X). Then for T ∈ B(X) a bounded linear operator, the spectrum of T, written σ(T), consists of those λ for which λ I - T is not invertible in B(X). Note that, when T is assumed to be closed, then by the closed graph theorem, this condition is equivalent to asserting that λ I - T fails to be bijective.
Classification of points in the spectrum of an operator
Loosely speaking, there are a variety of ways in which an operator T can fail to be invertible, and this allows us to classify the points of the spectrum into various types.
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, we necessarily have λ ∈ σ(T). The set of eigenvalues of T is sometimes called the point spectrum of T.
Approximate point spectrum
More generally, 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 x ∈ X. So the spectrum includes the set of approximate eigenvalues, which are those λ such that T - λ I is not bounded below; equivalently, it is the set of λ for which there is a sequence of unit vectors x1, x2, ... for which
- <math>\lim_{n \to \infty} \|Tx_n - \lambda x_n\| = 0</math>.
The set of approximate eigenvalues is known as the approximate point spectrum.
For example, in the example in the first paragraph of the bilateral shift on <math>\ell^2(\mathbf{Z})</math>, there are no eigenvectors, but every λ with |λ| = 1 is an approximate eigenvector; letting xn be the vector
- <math>\frac{1}\sqrt{n}(\dots, 0, 1, \lambda, \lambda^2, \dots, \lambda^{n-1}, 0, \dots)</math>
then ||xn|| = 1 for all n, but
- <math>Tx_n - \lambda x_n = \frac{2}\sqrt{n} \to 0</math>.
Compression spectrum
The unilateral shift on <math>\ell^2(\mathbf{N})</math> gives an example of yet another way in which an operator can fail to be invertible; this shift operator is bounded below (by 1; it is obviously norm-preserving) but it is not invertible as it is not surjective. The set of λ for which λ I - T is not surjective is known as the compression spectrum of T.
This exhausts the possibilities, since if T is surjective and bounded below, T is invertible.
Further results
If T is a compact operator, then it can be shown that any nonzero λ in the spectrum is an eigenvalue. In other words, the spectrum of such an operator, which was defined as a generalization of the concept of eigenvalues, consists in this case only of the usual eigenvalues, and possibly 0.
If X is a Hilbert space and T is 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).
Spectrum of unbounded operators
One can extend the definition of spectrum for unbounded operators on a Banach space X, operators which are no longer elements in the Banach algebra B(X). One proceeds in a manner similar to the bounded case. A complex number <math>\lambda</math> is said to be in the complement of the spectrum of a linear operator
- <math>T:\mathcal{D}\subset X\to X</math>
if the operator <math>T-\lambda I:\mathcal{D}\to X</math> is bijective, with its inverse a bounded operator. A complex number <math>\lambda</math> is then in the spectrum if this property fails to hold.
The spectrum of an unbounded operator is in general an unbounded subset of the complex plane. When dealing with closed unbounded operators, the closed graph theorem allows us to remove the boundedness requirement of the inverse. So for closed <math>T</math>, <math>\lambda </math> is in the spectrum of <math>T</math> iff <math>T-\lambda</math> is not bijective.
See also
References
- Dales et al, Introduction to Banach Algebras, Operators, and Harmonic Analysis, ISBN 0521535840de:Spektrum (Operatortheorie)
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