Much of the work in the previous two chapters has been preparatory, and in this chapter we arrive at some recent deep results, due to Ambrozie and Müller [8], concerning the existence of invariant subspaces for polynomially bounded operators on a complex Banach space with spectrum containing the unit circle. These results provide a powerful generalization of the celebrated results of Brown, Chevreau and Pearcy [53, 54], who proved the existence of invariant subspaces for contractions on a Hilbert space with spectrum containing the unit circle.

To follow this programme, it will be necessary to introduce a variety of themes that are of interest in their own right: Apostol sets, geometry of Banach spaces (in the form of Zenger's theorem) and Carleson interpolation. This chapter relies also on tools introduced in Chapters 2 and 3, namely, surjectivity of bilinear mappings and spectral measures.

We also emphasize the usefulness of a variety of functional calculi, such as those for holomorphic, C2(T) and H∞ functions. Further ways of using the functional calculus are presented in Chapter 5.

Apostol's theorem

We begin with a nice application of the holomorphic functional calculus, and a classical technique of integration through the spectrum.