Semi-weak n-hyponormality is defined and studied using the notion of positive determinant partition. Several examples related to semi-weakly n-hyponormal weighted shifts are discussed. In particular, it is proved that there exists a semi-weakly three-hyponormal weighted shift Wα with α0 = α1 < α2 which is not two-hyponormal, which illustrates the gaps between various weak subnormalities. Mathematics Subject Classification. 47B20, 47B37.Keywords. Hyponormal operators, quadratically hyponormal operators, polynomially hyponormal operators, weakly n-hyponormal operators.

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 Wα 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. Mathematics Subject Classification (2000). 47B37, 47B20.

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Abstract. It is known that a semi-cubically hyponormal weighted shift need not satisfy the flatness property, in which equality of two weights forces all or almost all weights to be equal. So it is a natural question to describe all semi-cubically hyponormal weighted shifts Wα with first two weights equal. Let α : 1, 1,∧ be a backward 3-step extension of a recursively generated weight sequence with 1 < x < u < v < w and let Wα be the associated weighted shift. In this paper we characterize completely the semicubical hyponormal Wα satisfying the additional assumption of the positive determinant coefficient property, which result is parallel to results for quadratic hyponormality.