In the previous chapter we saw that for every compact operator T on a Banach space X, the space can almost be written as a direct sum of generalized eigenspaces of T. If we assume that X is not merely a Banach space, but a Hilbert space, and T is not only compact but compact and normal, then such a decomposition is indeed possible – in fact, there is a decomposition with even better properties. Such a decomposition will be provided by the spectral theorem for compact normal operators: a complete and very simple description of compact normal operators. Thus with the study of a compact normal operator on a Hilbert space we arrive in the promised land: everything fits, everything works out beautifully, there are no blemishes. This is the best of all possible worlds.

We shall give two proofs of the spectral theorem, claiming the existence of the desired decomposition. In the first proof we shall make use of some substantial results from previous chapters, including one of the important results concerning the spectrum of a compact operator. The second proof is self-contained: we shall replace the results of the earlier chapters by easier direct arguments concerning Hilbert spaces and normal operators.

To start with, we collect a number of basic facts concerning normal operators in the following lemma. Most of these facts have already been proved, but for the sake of convenience we prove them again.

Lemma 1. Let T ∈ B(H) be a normal operator. Then the following assertions hold.