Motivated by the results of [6], we introduce a new class of operators, called \(\widehat{Q}\) -symmetric operators, defined via the \(*\) -Duggal transform. We provide a characterization of this class and study its basic properties. We also show that it contains several well-known classes of operators, such as quasinormal operators, idempotents, partial isometries, contractions, cyclic subnormal operators, and generalized quasi-adjoint operators. In the case of hyponormal operators, the generalized quasi-adjoint property is equivalent to the corresponding property of their Duggal transform. Furthermore, we prove several results concerning the ultraweak closures of the ranges of some elementary operators.
