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Existence of non-subnormal completely semi-weakly hyponormal weighted shifts

Li Chunji (Northeastern University, Department of Mathematics, Shenyang, P.R. China)
Lee Mi Ryeong (Catholic University of Daegu, Institute of Liberal Education, Gyeongsan, Gyeongbuk, Korea)

In this paper, we introduce a new notion of completely semi-weakly hyponormal operator which is a special case of polynomially hyponormal operator. For an one-step backward extension of the Bergman weighted shift, we show that completely semi-weakly hyponormal weighted shifts need not be subnormal. In addition, we provide an example which can serve to distinguish the semi-weak m-hyponormality from the semi-weak m-hyponormality with positive determinant coefficients for such a shift. Finally we discuss flatness on semi-weakly m-hyponormal weighted shifts.

Keywords: subnormal, polynomially hyponormal, semi-weakly hyponormal, weighted shifts