Schur product techniques for commuting multivariable weighted shifts

In this paper we study the hyponormality and subnormality of 2-variable weighted shifts using the Schur product techniques in matrices. As applications, we generalize the result in R. Curto, J. Yoon, Jointly hyponormal pairs of subnormal operators need not be jointly subnormal, Trans. Amer. Math. Soc. 358 (2006) 5135-5159, Theorem 5.2] and give a non-trivial, large class satisfying the Curto-Muhly-Xia conjecture R. Curto, P. Muhly, J. Xia, Hyponormal pairs of commuting operators, Oper. Theory Adv. Appl. 35 (1988) 1-22] for 2-variable weighted shifts. Further, we give a complete characterization of hyponormality and subnormality in the class of flat, contractive, 2-variable weighted shifts T≡(T₁,T₂) with the condition that the norm of the 0th horizontal 1-variable weighted shift of T is a given constant.