Completely Hyperexpansive Operator Tuples

The notion of a completely hyperexpansive operator on a Hilbert space is generalized to that of a completely hyperexpansive operator tuple, which in some sense turns out to be antithetical to the notion of a subnormal operator tuple with contractive coordinates. The countably many negativity conditions characterizing a completely hyperexpansive operator tuple are closely related to the Levy–Khinchin representation in the theory of harmonic analysis on semigroups. The interplay between the theories of positive and negative definite functions on semigroups forces interesting connections between the classes of subnormal and completely hyperexpansive operator tuples. Further, the several–variable generalization allows for a stimulating interaction with the multiparameter spectral theory.