The boundedness condition of dilation theory characterizes subnormals and contractions (Q806007)

B(H) denotes the set of all linear bounded operators in a complex Hilbert space H. Let S be a semigroup. A function A: \(S\to S\to B(H)\) satisfies boundedness condition (BC) if for each \(u\in S\) there is a nonnegative real number c(u) such that \[ \sum <A(us,ut)f(s),f(t)>\leq c(u)\sum <A(s,t)f(s),f(t)>,\quad \forall f\in F(S,H), \] where F(S,H) is the linear space of all functions from S to H vanishing off a finite subset of S. The author proves that subnormality of \(T\in B(H)\) is equivalent to BC with \(A(m,n)=T^{*n}T^ m\), \(S={\mathbb{N}}.\) The analogous characterization of the contractivity of a commutative semigroup homomorphism is obtained.