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Concept
Decomposition of spectrum (functional analysis)
Summary
The spectrum of a linear operator that operates on a Banach space is a fundamental concept of functional analysis. The spectrum consists of all scalars such that the operator does not have a bounded inverse on . The spectrum has a standard decomposition into three parts:
a point spectrum, consisting of the eigenvalues of ;
a continuous spectrum, consisting of the scalars that are not eigenvalues but make the range of a proper dense subset of the space;
a residual spectrum, consisting of all other scalars in the spectrum.
This decomposition is relevant to the study of differential equations, and has applications to many branches of science and engineering. A well-known example from quantum mechanics is the explanation for the discrete spectral lines and the continuous band in the light emitted by excited atoms of hydrogen.
Let X be a Banach space, B(X) the family of bounded operators on X, and T ∈ B(X). By definition, a complex number λ is in the spectrum of T, denoted σ(T), if T − λ does not have an inverse in B(X).
If T − λ is one-to-one and onto, i.e. bijective, then its inverse is bounded; this follows directly from the open mapping theorem of functional analysis. So, λ is in the spectrum of T if and only if T − λ is not one-to-one or not onto. One distinguishes three separate cases:
T − λ is not injective. That is, there exist two distinct elements x,y in X such that (T − λ)(x) = (T − λ)(y). Then z = x − y is a non-zero vector such that T(z) = λz. In other words, λ is an eigenvalue of T in the sense of linear algebra. In this case, λ is said to be in the point spectrum of T, denoted σp(T).
T − λ is injective, and its range is a dense subset R of X; but is not the whole of X. In other words, there exists some element x in X such that (T − λ)(y) can be as close to x as desired, with y in X; but is never equal to x. It can be proved that, in this case, T − λ is not bounded below (i.e. it sends far apart elements of X too close together).
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