The Lifting Problem for Hyponormal Pairs of Commuting Subnormal Operators

We construct three different families of commuting pairs of subnormal operators, jointly hyponormal but not admitting commuting normal extensions. Each such family can be used to answer in the negative a 1988 conjecture of RC, P. Muhly and J. Xia. We also obtain a sufficient condition under which joint hyponormality does imply joint subnormality. Our tools include the use of 2-variable weighted shifts, the six-point test for joint hyponormality, disintegration of measures techniques, the theory of multivariable moment problems, and matrix positivity. We obtain new necessary conditions for the existence of a lifting, and generate new pathology associated with bringing together the Berger measures associated to each individual weighted shift. For subnormal 2-variable weighted shifts, we then find the precise relationship between the Berger measure of the pair and the Berger measures of the shifts associated to horizontal rows and vertical columns of weights. Finally, we consider the (multivariable) spectral theory of these hyponormal pairs, and discover some unexpected new phenomena, not present in the single variable theory. [email protected]; http://www.math.uiowa.edu/ ̃rcurto