SEVERAL GRADED CONTRACTIONS ON HILBERT-SPACE

In this paper we define the concept of amplification for several commuting contractions on Hilbert space. The question of symbolic calculus for such contractions is addressed and reduced to a special case in which the contractions are independent unilateral shifts at least on the algebraic level. Using amplifications, this question is further reduced to the case in which the underlying "geometry" of the Hilbert space is "maximal". In the case of one or two contractions, these maximal geometries correspond to the case in which the contractions are isometries. For three or more contractions, the structure of maximal geometries remains a mystery. Based on this approach, we prove some new results on symbolic calculus-generalizations of the inequalities of von Neumann and Ando. The most interesting result is for homogenous polynomials whose spectrum is "centrally located". Some of these ideas apply with routine modifications to the case where the contractions do not necessarily commute. We also discuss cases where some commutations are allowed, but not others. © 1990.