Recursiveness and propagation for 2-variable weighted shifts

For 1-variable weighted shifts, it is well known that given three positive numbers α0<α1<α2, we always find Stampfli's subnormal weighted shift whose first three weights are α0, α1 and α2 (we call it a type-abc completion, or shift.) On the other hand, given four positive numbers α0<α1<α2<α3, it is not always possible to find a subnormal weighted shift whose first four weights are α0, α1, α2 and α3. 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.