A complex number λ is an extended eigenvalue of an operator A if there is a nonzero operator B such that AB = λBA. In this case, B is said to be an eigenoperator. This research paper is devoted to the investigation of some results of extended eigenvalues for a closed linear operator on a complex Banach space. The obtained results are explored in terms two cases bounded, and closed eigenoperators. In addition, the notion of extended eigenvalues for a 2 × 2 upper triangular operator matrix is introduced and some of its properties are displayed.