Decomposition of spectrum (functional analysis)

In mathematics, especially functional analysis, the spectrum of an operator generalizes the notion of eigenvalues. Given an operator, it is sometimes useful to break up the spectrum into various parts. This article discusses a few examples of such decompositions.

Operators on Banach space

Let X be a Banach space, L(X) the family of bounded operators on X.

A complex number λ is in the spectrum of T, denoted σ(T), if T − λ does not have bounded inverse. If T − λ is invertible (i.e., if it is one-to-one and onto), then its inverse is bounded; this follows directly from the open mapping theorem. So, λ is in the spectrum of T if and only if T − λ is either not one-to-one or not onto. One can easily check that if T − λ is one-to-one, bounded below (i.e. does not send far apart elements of X too close together), and has dense range, then in fact T − λ must be onto, so λ will not be in σ(T). Therefore, if λ is in σ(T), one of the following must be true:

1. T − λ is not injective.
2. T − λ is injective, and has dense range. But T − λ is not bounded below, and we fail to get that the range is all of X. Equivalently, the densely-defined linear map (T − λ) x → x is not bounded, therefore can not be extended to all of......

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