In this chapter we apply the results of Chapter 9 to the study of multiply connected K-spectral sets. We show that for a “nice” region X with finitely many holes it is possible to write down a fairly simple characterization of the family of operators that, up to similarity, have normal ∂X-dilations. This constitutes a model theory for these operators. In contrast, if X has two or more holes, then it is still an open problem to determine whether or not every operator for which X is a spectral set has a normal ∂X-dilation, i.e., is a complete spectral set. A further difficulty with the theory of spectral sets is that it is quite difficult to determine if a given set is a spectral set for an operator. We will illustrate this difficulty in the case that X is an annulus and T is a 2 × 2 matrix.

Thus, even if it is eventually determined that the properties of being a spectral set and being a complete spectral set are equivalent, the use of the theory might be limited by the impossibility of recognizing operators to which it could be applied.

It is easier to determine when a “nice” set with no holes is a spectral set for an operator.