Abstract.
The gap between hyponormal and subnormal Hilbert space operators can be studied using the intermediate classes of weakly n-hyponormal and (strongly) n-hyponormal operators. The main examples for these various classes, particularly to distinguish them, have been the weighted shifts. In this paper we first obtain a characterization for a weakly n-hyponormal weighted shift W α with weight sequence α, from which we extend some known results for quadratically hyponormal (i.e., weakly 2-hyponormal) weighted shifts to weakly n-hyponormal weighted shifts. In addition, we discuss some new examples for weakly n-hyponormal weighted shifts; one illustrates the differences among the classes of 2-hyponormal, quadratically hyponormal, and positively quadratically hyponormal operators.
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Exner, G., Jung, I.B. & Park, S.S. Weakly n-hyponormal Weighted Shifts and Their Examples. Integr. equ. oper. theory 54, 215–233 (2006). https://doi.org/10.1007/s00020-004-1360-2
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DOI: https://doi.org/10.1007/s00020-004-1360-2