A new approach to the 2-variable subnormal completion problem

We study the Subnormal Completion Problem (SCP) for 2-variable weighted shifts. We use tools and techniques from the theory of truncated moment problems to give a general strategy to solve SCP. We then show that when all quadratic moments are known (equivalently, when the initial segment of weights consists of five independent data points), the natural necessary conditions for the existence of a subnormal completion are also sufficient. To calculate explicitly the associated Berger measure, we compute the algebraic variety of the associated truncated moment problem; it turns out that this algebraic variety is precisely the support of the Berger measure of the subnormal completion

Hyponormality and subnormality for powers of commuting pairs of subnormal operators

Let H_0 (resp. H_\infty denote the class of commuting pairs of subnormal operators on Hilbert space (resp. subnormal pairs), and for an integer k>=1 let H_k denote the class of k-hyponormal pairs in H_0. We study the hyponormality and subnormality of powers of pairs in H_k. We first show that if (T_1,T_2) is in H_1, then the pair (T_1^2,T_2) may fail to be in H_1. Conversely, we find a pair (T_1,T_2) in H_0 such that (T_1^2,T_2) is in H_1 but (T_1,T_2) is not. Next, we show that there exists a pair (T_1,T_2) in H_1 such that T_1^mT_2^n is subnormal (all m,n >= 1), but (T_1,T_2) is not in H_\infty; this further stretches the gap between the classes H_1 and H_\infty. Finally, we prove that there exists a large class of 2-variable weighted shifts (T_1,T_2) (namely those pairs in H_0 whose cores are of tensor form) for which the subnormality of (T_1^2,T_2) and (T_1,T_2^2) does imply the subnormality of (T_1,T_2)

k-hyponormality of multivariable weighted shifts

We characterize joint k-hyponormality for 2-variable weighted shifts. Using this characterization we construct a family of examples which establishes and illustrates the gap between k-hyponormality and (k+1)-hyponormality for each k>=1. As a consequence, we obtain an abstract solution to the Lifting Problem for Commuting Subnormals.Comment: 13 pages; to appear in J. Funct. Ana