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. 
Raul E. Curto, Sang Hoon Lee and Jasang Yoon