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.
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Poon, Y.T., Yoon, J. Quadratically Hyponormal Recursively Generated Weighted Shifts Need Not Be Positively Quadratically Hyponormal. Integr. equ. oper. theory 58, 551–562 (2007). https://doi.org/10.1007/s00020-007-1505-1
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DOI: https://doi.org/10.1007/s00020-007-1505-1