Extended eigenvalues of 2 × 2 block operator matrices

Abstract

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