Existence of Non-subnormal Completely Semi-Weakly Hyponormal Weighted Shifts

Abstract

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.