Abstract: We considered  the relations between the property (UW_Π) and the hypercyclic property for a bounded linear operator or operator  functions. By defining the new spectral set, we gave a  judgement method for  bounded linear operators or operator  functions satisfying property (UW_Π) and hypercyclic property at the same time, and the obtained results were applied.

Key words:  bounded linear operator, property (UW_Π), hypercyclic property, spectrum