Berger Measure For S(a,b,c,d) And Subnormal Backward Extension Theorem

In this note, we get the Berger measure of the weighted shift S(a, b, c, d) with weights σn:=√an-b/cn+d(a,b,c,d>0and n≥0) as well as its p-subshift. Then we will give ex-amples from analytic function theory to illustrate that the necessary condition given by Curto and Yoon (2006)[14] for sub normality of2-variable weighted shift is not suffi-cient. Furthermore, we will do some calculations about subnormal backward extension of S(a, b, c, d) with the Berger measure of S(a, b, c, d) which has been gotten. Finally, we use the subnormal backward extension of a2-variable weighted shift to prove that if2-variable weighted shifts T=(T1,T2)∈TC(see definition4.1), the subnormality of (T1,Tn2),(Tn1,T2) and(T1,T2) are equivalent, and then we will get the subnormality of (T1,T2) and (Tm1,Tn2) are equivalent, which shows a method that is straightforward and different from [11].