Flat phenomena of 2-variable weighted shifts

We study the class of mono-weakly 2-hyponormal 2-variable weighted shifts T≡(T1,T2)T≡(T1,T2) (resp. hyponormal with quadratically hyponormal T1T1 and T2T2) with two consecutive equal weights in the weight sequence of each of the coordinate operators. We show that under natural assumptions on the coordinate operators, the presence of consecutive equal weights in T1T1 and T2T2 leads to the flatness, in a way that resembles the situation for 1-variable weighted shifts.