Spectrum of an operator

In functional analysis, the concept of the spectrum of an operator is a generalisation of the concept of eigenvalues, which is much more useful in the case of operators on infinite-dimensional spaces. For example, the bilateral shift operator on the Hilbert space <math>\ell^2(\mathbb{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.

Definition

Let X be a complex Banach space, and B(X) the Banach algebra of bounded linear operators on X. Then if I denotes the identity operator, and T ∈ B(X) then the spectrum of T (normally written as σ(T) ) consists of λ such that λ I - T is not invertible in the algebra of bounded linear operators on X. Note that by the closed graph theorem, this condition is equivalent to asserting λ I - T fails to be bijective.

Basic properties

Theorem: The spectrum is non-empty, bounded, and closed.

Proof: Suppose the spectrum is empty; then the function R(λ) = (λI - T)^-1 is defined everywhere on the complex plane. So if Φ is any linear functional on B(X), F(λ) = Φ(R(λ)) is a continuous function <math>\mathbb{C}\to\mathbb{C}<math>. It is not hard to see that

<math>\lim_{\mu \to \lambda} \frac{F(\lambda) - F(\mu)}{\lambda - \mu} = -\Phi( R(\lambda)^2 )<math>

so F is an analytic function. However, F(λ) is O(λ^-1) for large λ so F is a bounded analytic function, and hence constant by Liouville's theorem, and thus everywhere zero as it is zero at infinity. However, by the Hahn-Banach theorem this implies that R(λ) is zero for all λ, which is obviously a contradiction.

The boundedness of the spectrum is immediate from the Neumann series expansion (named after the German mathematician Carl Neumann),

<math>(I - A)^{-1} = \sum_{n = 0}^\infty A^n<math>,

which is valid for any A ∈ B(X) with ||A|| < 1. This implies that if |λ| > ||T||, (λ I - T) is invertible (taking A = T/λ). So σ(T) is bounded, and the spectral radius

<math>r(T) = \sup \{|\lambda| : \lambda \in \sigma(T)\}<math>

is bounded above by ||T||.

Furthermore, the Neumann series implies that for any two operators A, B with A invertible and ||A - B|| < ||A^-1||^-1, B must also be invertible. So the set of invertible operators is open, and hence, since the function <math>\mathbb{C}\to B(X)<math> defined by <math>\lambda \mapsto \lambda I - T<math> is continuous, the set of λ for which λ I - T is invertible is open, so its complement is closed; but this complement is exactly σ(T).

Classification of points in the spectrum

Loosely speaking, there are a variety of ways in which an operator S 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 S(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, S is not invertible if it is not bounded below; that is, if there is no 'c' > 0 such that ||Sx|| > 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(\mathbb{Z})<math>, there are no eigenvectors, but every λ with |λ| = 1 is an approximate eigenvector; letting x_n be the vector

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

then ||x_n|| = 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(\mathbb{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

The spectral radius formula states that

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

This can be proved using similar methods to the above theorem, considering the power series expansion of F(1/λ); this must converge for all λ > r(T), and applying the uniform boundedness principle to the series coefficients gives the result.

If T is a compact operator, then it can be shown that any nonzero approximate eigenvalue is in fact an eigenvalue.

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).