SOME MAPPING THEOREMS FOR THE CLASSES ${\Bbb A}_{n, m} $ AND THE BOUNDARY SETS

Abstract

Denote by $\Bbb A$ the class of all absolutely continuous contractions whose associated Sz. Nagy-Foias functional calculus is isometric. Starting from the fact that if $u$ is a non-constant inner function and if $T\in {\Bbb A}$, then so does $u(T)$, we study how inner functions operate on the classes ${\Bbb A}_{n,m}$, subclasses of the class ${\Bbb A}$. For this purpose, we use standard dual algebra techniques and a decomposition of the algebra $H^\infty$ into a weak*-topological direct sum of copies of itself. We also discuss mapping theorems for the support of the spectral measures associated with the unitary parts of the minimal isometric extension and the minimal co-isometric extension of an absolutely continuous contraction $T$.