and subnormal safe quotients for geometrically regular weighted shifts

Geometrically regular weighted shifts (in short, GRWS) are those with weights α(N,D) given by αn(N,D)=pn+Npn+D, where p>1 and (N,D) is fixed in the open unit square (−1,1)×(−1,1). We study here the zone of pairs (M,P) for which the weight α(N,D)α(M,P) gives rise to a moment infinitely divisible (▪) or a subnormal weighted shift, and deduce immediately the analogous results for product weights α(N,D)α(M,P), instead of quotients. Useful tools introduced for this study are a pair of partial orders on the GRWS.