GKS Decomposition and Spherical Dilations

Using the generalized version of the classical F. and M. Riesz Theorem as given by Gliksberg, Kِnig, and Seever, we obtain a few decomposition theorems for tuples of commuting operators on Hilbert spaces that admit normal dilations whose joint spectra are contained in the unit sphere of Cn. Our results apply in particular to sphericaln-hypercontractions, subnormaln-tuples whose joint spectra are contained in the closed unit ball of Cn, and to spherical isometries. The questions related to the uniqueness of decompositions are addressed by appealing to a specialized version of an approximation result related to the solution of the inner function problem on the unit ball of Cn. The Henkin measures on the unit sphere play a central role in the development of the relevant theory.