Schur product techniques for the subnormality of commuting 2-variable weighted shifts

In this paper, we study the subnormality of 2-variable weighted shifts using the Schur product techniques in matrices and the convolution of two continuous functions. As a consequence, we find the Berger measure of the subnormal weighted shift obtained from the Schur product of two subnormal weighted shifts. As applications, we first give non-trivial, large classes satisfying the Curto-Muhly-Xia conjecture (see the conjecture given below) for 2-variable weighted shifts. We next show when the 2-hyponormality of 2-variable weighted shifts becomes subnormality.