4 - Polynomially bounded operators with rich spectrum

Summary

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.