Quadratically hyponormal weighted shifts with two equal weights

We present an extensive analysis of positively quadratically hyponormal wei ghted shifts W-alpha with alpha(0) = alpha(1) = 1. Our main result states t hat such weighted shifts abound! Specifically, by focusing on recursively g enerated weighted shifts of the form W-1,W-(1,W-root x,W-root y)boolean AND , we establish that the planar set R := {(x, y) : W-1,W-(1,W-root x,W-root y)boolean AND is positively quadratically hyponormal} is a closed convex se t with nonempty interior. In addition, we are able to describe in detail th e boundary of R.