Quadratically hyponormal recursively generated weighted shifts need not be positively quadratically hyponormal

Abstract

We study a class of weighted shifts W α defined by a recursively generated sequence α ≡ α0, ... , α m-2, (α m-1, α m , α m+1)∧ and characterize the difference between quadratic hyponormality and positive quadratic hyponormality. We show that a shift in this class is positively quadratically hyponormal if and only if it is quadratically hyponormal and satisfies a finite number of conditions. Using this characterization, we give a new proof of [12, Theorem 4.6], that is, for m = 2, W α is quadratically hyponormal if and only if it is positively quadratically hyponormal. Also, we give some new conditions for quadratic hyponormality of recursively generated weighted shift W α (m ≥ 2). Finally, we give an example to show that for m ≥ 3, a quadratically hyponormal recursively generated weighted shift W α need not be positively quadratically hyponormal. © 2007 Birkhäuser Verlag Basel/Switzerland.