Spectrum (functional analysis)

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In functional analysis, the concept of the spectrum of an operator is a generalisation of the concept of eigenvalues for matrices. Operators on infinite-dimensional spaces may have no eigenvalues. For example, on the Hilbert space ℓ2 the unilateral shift operator,

(x_1, x_2, \dots) \mapsto (0, x_1, x_2, \dots),

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.

A bounded operator may be viewed as an element of a Banach algebra, with the definition of spectrum transferred verbatim from that context. The notion of spectrum extends to unbounded operators. In the unbounded case, often the operator is required to be closed in order to obtain nice spectral properties.

The study of spectra and related properties is known as spectral theory.

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[edit] Spectrum of a bounded operator

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.

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). The spectrum of a bounded linear operator is its spectrum when 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). In particular, every bounded operator is closed. Therefore by the closed graph theorem, if λ I − T is bijective then λ does not lie in the spectrum of T.

[edit] 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(\lambda) = (\lambda I - x)^{-1} \,

would be defined everywhere on the complex plane and bounded. But it can be shown that the resolvent function R 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 in λ; the spectrum σ(x) is bounded by ||x||. A similar result shows that the closedness of the spectrum and hence the spectrum of a bounded operator is compact.

The bound ||x|| on the spectrum can be refined somewhat. The spectral radius, r(x), of x is 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, i.e.

r(x) = \sup \{|\lambda| : \lambda \in \sigma(x)\}.

The spectral radius formula says that

r(x) = \lim_{n \to \infty} \|x^n\|^{1/n}.

[edit] Classification of points in the spectrum of an operator

For a bounded operator T, T is invertible, i.e. has a bounded inverse, if and only if T is bounded below and has dense range. Accordingly, the spectrum of T can be divided into the following parts:

  1. If λ ∈ σ(T), it may be that λ - T is not bounded below. Since T is bounded, for every eigenvalue λ of T, λ - T is not bounded below. The set of eigenvalues is called the point spectrum of T. Alternatively, λ - T could be one-to-one but still not be bounded below. Such λ is said to be in the approximate point spectrum of T.
  2. Or, λ - T may not have dense range. In that case, λ is said to be in the residual spectrum of T

Notice that, in principle, bijectivity is sufficient but not necessary for invertibility. Sufficiency is due to the closed graph theorem. Also, defined in this way, the parts of the spectrum need not be disjoint.

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

[edit] 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 also called the point spectrum of T.

[edit] 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

\lim_{n \to \infty} \|Tx_n - \lambda x_n\| = 0.

The set of approximate eigenvalues is known as the approximate point spectrum.

When T is bounded, then, by Riesz's lemma, the eigenvalues lie in the approximate point spectrum.

Example Consider the bilateral shift T on l2(Z) defined by

T(\cdots, a_{-1}, \hat{a}_0, a_1, \cdots) = (\cdots, \hat{a}_{-1}, a_0, a_1, \cdots)

where the ˆ denotes the zero-th position. Direct calculation shows T has no eigenvalues, but every λ with |λ| = 1 is an approximate eigenvalue; letting xn be the vector

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

then ||xn|| = 1 for all n, but

\|Tx_n - \lambda x_n\| = \sqrt{\frac{2}{n}} \to 0.

Since T is a unitary operator, its spectrum lie on the unit circle. Therefore the approximate point spectrum of T is its entire spectrum. This is true for a more general class of operators.

A unitary operator is normal. By spectral theorem, a bounded operator on a Hilbert space is normal if and only if it is a multiplication operator. It can be shown that, in general, the approximate point spectrum of a bounded multiplication operator is its spectrum.

When T is unbounded, the definition of approximate point spectrum is slightly different. Continuity can no longer be used to show that that every eigenvalue is an approximate eigenvalue. So the approximate point spectrum of T is defined to be the union of eigenvalues and approximate eigenvalues.

[edit] Residual spectrum

An operator may be bounded below but not invertible. The unilateral shift on l 2(N) is such an example. This shift operator is an isometry, therefore bounded below by 1. But it is not invertible as it is not surjective. The set of λ for which λ I - T does not have dense range is known as the residual spectrum or compression spectrum of T.

[edit] 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).

[edit] 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 λ is said to be in the resolvent set, that is, the complement of the spectrum of a linear operator

T: D \subset X \to X

if the operator

T-\lambda I: D \to X

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

S : X \rightarrow D

such that

S (T - \lambda) = I_D, \, (T - \lambda) S = I_X.

A complex number λ is then in the spectrum if this property fails to hold. One can classify the spectrum in exactly the same way as in the bounded case.

The spectrum of an unbounded operator is in general a closed, possibly empty, subset of the complex plane.

Immediately from the definition, it can be deduced that S can not be invertible, in the sense of bounded operators. Since the domain D may be a proper subset of X, the expression

\, (T - \lambda) S = I_X

makes sense only if Ran(S) is contained in D. Similarly,

\, S (T - \lambda) = I_D

implies D ⊂ Ran(S). Therefore, λ being in the resolvent set of T means

T-\lambda I: D \to X

is bijective. (Recall that bijectivity of T - λ is not implied by invertibility if T is bounded.)

The converse is true if one introduces the additional assumption that T is closed. By the closed graph theorem, if T - λ: D → X is bijective, then its (algebraic) inverse map is necessarily a bounded operator. (Notice the completeness of X is required in invoking the closed graph theorem.) Therefore, in contrast to the bounded case, the condition that a complex number λ lie in the spectrum of T becomes a purely algebraic one: for a closed T, λ is in the spectrum of T if and only if T − λ is not bijective.

[edit] See also

[edit] References

  • Dales et al, Introduction to Banach Algebras, Operators, and Harmonic Analysis, ISBN 0-521-53584-0