Subnormality of Bergman-like weighted shifts

For a,b,c,d⩾0 with ad−bc>0, we consider the unilateral weighted shift S(a,b,c,d) with weights αn:=an+bcn+d(n⩾0). Using Schur product techniques, we prove that S(a,b,c,d) is always subnormal; more generally, we establish that for every p⩾1, all p-subshifts of S(a,b,c,d) are subnormal. As a consequence, we show that all Bergman-like weighted shifts are subnormal.