Using the generalized version of the classical F. and M. Riesz Theorem as given by Gliksberg, Ko nig, 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 C n . Our results apply in particular to spherical n-hypercontractions, subnormal n-tuples whose joint spectra are contained in the closed unit ball of C n , 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 C n . The Henkin measures on the unit sphere play a central role in the development of the relevant theory.
1998 Academic Press
1. FUNCTION THEORETIC PRELIMINARIES
If X is a compact Hausdorff space, then C(X) will denote the algebra of complex-valued continuous functions on X with the sup norm. A function algebra on X is a subalgebra of C(X) which is closed in the sup norm (uniform) topology, which contains constants, and which separates points on X. The set of regular complex Borel measures on X is denoted by M(X) and is identified with the dual C(X)* of C(X). If A is a function algebra on X and 8 is a multiplicative linear functional on A, then M 8 denotes the set of all probability measures \ in M(X) that represent 8 in the sense that
Note that M 8 is a convex subset of M(X) and is also weak*-compact. If \ in M(X) is such that X f d=0 for every f in A, then we write =A. Crucial for our purposes is the following decomposition theorem due to Gliksberg [G] and Ko nig and Seever [KS].