We continue in this lecture to consider dilation theory of contractions. In the theory of Sz.-Nagy and Foias, the main route proceeded from the existence of the minimal isometric and unitary dilations, to the study of how such dilations look like, and to use them to extract information about an operator from its dilations. The only application we saw until now was von Neumann’s inequality, which is not a trivial fact, but let’s admit it: somewhat rinky dinky. But after a deeper look is taken into the structure of the minimal unitary dilation, the way for more significant applications opens. One of these applications is a functional calculus (for non-selfadjoints) that extends the holomorphic functional calculus. Another application is an affirmative solution to the invariant subspace problem for certain classes of operators. The main parts of this theory are laid down in the book Harmonic Analysis of Operators On Hilbert Space.
We will not follow that route. Rather, we will see what dilation theory can help us to understand regarding tuples of commuting operators (which is also treated to some extent in the book). Surprisingly, this will lead to a truly nifty application in function theory. Read the rest of this entry »