Recursiveness and propagation for 2-variable weighted shifts

摘要

For 1-variable weighted shifts, it is well known that given three positive numbers alpha(0) < alpha(1) < alpha(2), we always find Stampfli's subnormal weighted shift whose first three weights are alpha(0), alpha(1) and alpha(2) (we call it a type-abc completion, or shift.) On the other hand, given four positive numbers alpha(0) < alpha(1) < alpha(2), it is not always possible to find a subnormal weighted shift whose first four weights are alpha(0), alpha(1), alpha(2) and alpha(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. (C) 2016 Elsevier Inc. All rights reserved.