Recursiveness and propagation for 2-variable weighted shifts.

For 1-variable weighted shifts, it is well known that given three positive numbers α 0 < α 1 < α 2 , we always find Stampfli's subnormal weighted shift whose first three weights are α 0 , α 1 and α 2 (we call it a type-abc completion, or shift .) On the other hand, given four positive numbers α 0 < α 1 < α 2 < α 3 , it is not always possible to find a subnormal weighted shift whose first four weights are α 0 , α 1 , α 2 and α 3 . In this paper, we extend the notion of a type- abc shift to the 2-variable weighted shifts using a localization technique from the theory of truncated moment problems. We next study an inner propagation phenomena of a 2-variable weighted shift whose core is of type- abc . We further establish the necessary and sufficient conditions for the subnormality of one-step extensions of type- abc 2-variable weighted shifts. Finally, we study a certain invariant property under powers for type-abc 2-variable weighted shifts. [ABSTRACT FROM AUTHOR]