EXTREMAL PROBLEM OF A QUADRATICALLY HYPONORMAL WEIGHTED SHIFT

Let $W_{\alpha}$, be a recursively generated quadratically hyponormal weighted shift with a weight sequence ${\alpha}$ : 1, (1, $\sqrt{x}$, $\sqrt{y}$)$^{\wedge}$. In [4] Curto-Jung showed that R = {(x,y) : $W_{1,\;(1,\;{\sqrt{x}},\;{\sqrt{y}})^{\wedge}}$ is quadratically hyponormal} is a closed convex with nonempty interior, which guarantees that there are a lot of quadratically hyponormal weighted shifts with first two equal weights. They suggested a problem computing expressions of certain extremal points of R. In this note we obtain a partial answer of their extremal problem.