A moment theorem for contractions on Hilbert spaces (Q579598)

Let X be a subset of a Hilbert space H which spans H, and let f be a function from \(Z\times X\) to H, where Z denotes the set of integers. This paper is concerned with the following moment problem: When does there exist a contraction T on H such that \(T_ nx=f(n,x)\) for all \(n\in Z\) and \(x\in X\), where \(T_ n\) is defined as \[ T_ n=T^ n\;if\;n\geq 0\quad and\quad =T^{*^{| n|}}\;if\;n<0. \] This note answers this question completely by showing that some natural necessary conditions are also sufficient. The proof for the necessity makes use of the (power) unitary dilation of the contraction T. A continuous analogue of this problem is also considered.