GKS decomposition and spherical dilations (Q1270222)

Generalizations of the classical F. and M. Riesz theorem have been given by \textit{I. L. Glicksberg} [J. Funct. Anal. 1, 109-122 (1967; Zbl 0155.18503)] and \textit{H. König} and \textit{G. L. Seever} [Duke Math. J. 36, 791-797 (1969; Zbl 0201.45704)]. In the present paper these results are applied to obtain several decomposition theorems for \(n\)-tuples of commuting operators on Hilbert space that admit normal dilations whose joint spectra are contained in the unit sphere of \(\mathbb{C}^n\). The results obtained apply to spherical \(n\)-hypercontractions, \(n\)-tuples of subnormal operators whose joint spectra are contained in the closed unit ball of \(\mathbb{C}^n\), and to spherical isometries. The uniqueness or otherwise of the decompositions is discussed.