Sebestyén Moment Problem: the Multi-dimensional Case

Given a family {hn}n∈ZΩ+ of vectors in a Hilbert space H we characterize the existence of a family of commuting contractions T = {Tω}w∈Ω on H having regular dilation and such that hn = T h0, n ∈ Z+. The theorem is a multi-dimensional analogue for some well-known operator moment problems due to Sebestyén in case |Ω| = 1 or, recently, to Găvruţă and Păunescu in case |Ω| = 2. Let H be a complex Hilbert space. Denote by L(H) the C∗-algebra of bounded linear operators on H. Z. Sebestyén proposed in [5] the following operator moment problem: Problem A. Given a sequence {hn}n≥0 of vectors in H, under what condition does there exist a contraction T on H such that hn = T h0, n ≥ 0 ? The two-dimensional analogue of this problem was proposed by Găvruţă and Păunescu [2]: Problem B. Given a doubly indexed sequence {hm}m,n≥0 of vectors in H, under what condition does there exist a commuting pair of contractions (S, T ) on H such that hm = S T h0, m, n ≥ 0 ? The solution given by Z. Sebestyén to the first problem was the following: Theorem A ([5]). Let {hn}n≥0 ⊂ H. Problem A has a solution if and only if ‖ ∑ m,m′ cm,m′hm+m′‖ ≤ ∑ m,m′,n,n′ cm,m′cn,n′〈h(m−n)++m′ , h(m−n)−+n′〉, for every finite double sequence {cn,n′}n,n′≥0 of complex numbers (we used, for any integer m, the notation m = max{m, 0} and m− = −min{m, 0}). Received by the editors October 22, 2002. 2000 Mathematics Subject Classification. Primary 47A57, 47A20.