Spectrum: Information from Answers.com

In functional analysis, the concept of the spectrum of a bounded operator is a generalisation of the concept of eigenvalues for matrices. Specifically, a complex number λ is said to be in the spectrum of a bounded linear operator T if λI − T is not invertible, where I is the identity operator. 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,

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

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 the operator R − 0 (i.e. R itself) is not invertible: it is not surjective since any vector with non-zero first component is not in its range. In fact every bounded linear operator on a complex Banach space must have a non-empty spectrum.

The notion of spectrum extends to densely-defined unbounded operators. In this case a complex number λ is said to be in the spectrum of such an operator T:D→X (where D is dense in X) if there is no bounded inverse (λI − T)^-1:X→D. If T is a closed operator (which includes the case that T is a bounded operator), boundedness of such inverses follow automatically if the inverse exists at all.

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.

1 Spectrum of a bounded operator
- 1.1 Basic properties
2 Classification of points in the spectrum of an operator
3 Further results
4 Spectrum of an unbounded operator
5 Spectrum of a unital Banach algebra
6 See also
7 References

Spectrum of a bounded operator

The spectrum of a bounded linear operator T acting on a Banach space X is the set of complex numbers λ such that λI − T does not have an inverse that is a bounded linear operator. If λI − T is invertible then that inverse is linear (this follows immediately from the linearity of λI − T), and by the bounded inverse theorem is bounded. Therefore the spectrum consists precisely of those λ where λI − T is not bijective.

The spectrum of a given operator T is denoted σ(T), and the resolvent set (the set of points not in the spectrum) is denoted ρ(T).

Basic properties

The spectrum of a bounded operator T is always a closed, bounded and non-empty subset of the complex plane.

If the spectrum were empty, then the resolvent function

$R(\lambda) = (\lambda I - T)^{-1} \,$

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(T) is bounded by ||T||. A similar result shows the closedness of the spectrum.

The bound ||T|| on the spectrum can be refined somewhat. The spectral radius, r(T), of T is the radius of the smallest circle in the complex plane which is centered at the origin and contains the spectrum σ(T) inside of it, i.e.

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

The spectral radius formula says that

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

Classification of points in the spectrum of an operator

Further information: Decomposition of spectrum (functional analysis)

This section's factual accuracy is disputed. Please see the relevant discussion on the talk page. (June 2009)

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:

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.
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 bounded inverse 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.

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.

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 x_n be the vector

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

then ||x_n|| = 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 every eigenvalue is an approximate eigenvalue. So the approximate point spectrum of T is defined to be the union of eigenvalues and approximate eigenvalues.

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 λI - T does not have dense range is known as the residual spectrum or compression spectrum of T.

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 an unbounded operator

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.

For λ to be in the resolvent (i.e. not in the spectrum), as in the bounded case λI − T 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.

However, boundedness of the inverse does follow directly from its existence if one introduces the additional assumption that T is closed; this follows from the closed graph theorem. Therefore, as in the bounded case, a complex number λ lies in the spectrum of a closed operator T if and only if λI − T is not bijective. Note that the class of closed operators includes all bounded operators.

Spectrum of a unital Banach algebra

This section requires expansion.

Let B be a complex Banach algebra containing a unit e. Then we define the spectrum σ(x) (or more explicitly σ_B(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 Banach algebra.

References

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

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