Operator Interpolation and the Commutant

Abstract

The construction and the study of a rich functional calculus is one way of getting spectral resolutions or their analogues. In the classical case of a self-adjoint (or normal) operator A the calculus f ↦ f(A) defined on the algebra of all bounded functions and, in particular, on the characteristic functions χ_σ generates a projection valued (spectral) measure $\sigma \mapsto E\left( \sigma \right)\mathop = \limits^{def} {\chi _\sigma }\left( A \right)$, the study of which leads to the Spectral Theorem. This measure “separates” the spectrum of A sufficiently well so that, for instance, if A is complete we obtain an unconditionally convergent expansion into eigenvectors. For more general transformations having a total and topologically free family X of eigen (or root) spaces one can always define an analogue of the spectral measure, in a similar manner as this is provided by formula (6), Lecture VI, but in the first instance only on the (dense) linear hull of the family X. If moreover the eigenvalues of the operator are isolated points of the spectrum then this “measure” can be considered as a part of the calculus (the “analytical” Riesz-Dunford calculus). For the compressions of the shift operator T = PS∣K we have to use the calculus constructed in Lecture III beyond which (as will be shown) we cannot go. The Sz.-Nagy and Foiaş calculus, it is true, makes it possible to construct the operator f(T) only for analytic functions f (f ∈ H^∞) but this deficiency is compensated by the fact that in many cases f(T) depends only on f ∣σ(T) or even f ∣σ _p (T). Such functions already can separate the spectrum of T in the desired way.