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 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) = (λ*I* - *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 **C****C**. It is not hard to see that

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),

- ,

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*

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. It follows that the set of invertible operators is open, and hence, since the function **C** → *B*(*X*) defined by λ → λ *I* - *T* 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 *x*_{1}, *x*_{2}, ... for which

- .

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 , there are no eigenvectors, but every λ with |λ| = 1 is an approximate eigenvector; letting *x*_{n} be the vector

then ||*x*_{n}|| = 1 for all *n*, but

- .

### Compression spectrum

The unilateral shift on 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

- .

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).

## See also

## External link

An account of the spectral theorem

Last updated: 06-01-2005 21:28:32