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 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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2 Basic properties 3 Classification of points in the spectrum 4 Further results 5 External link |
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) ) is defined by
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 . It is not hard to see that
Basic properties
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 von Neumann series expansion,
- ,
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 defined by 
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).
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.
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 xn be the vector
This exhausts the possibilities, since if T is surjective and bounded below, T is invertible.
The spectral radius formula states that
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). Classification of points in the spectrum
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
The set of approximate eigenvalues is known as the approximate point spectrum.
then ||xn|| = 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.
Further results
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.