A class of operators related to m-symmetric operators

Abstract

m-symmetric operator plays a crucial role in the development of operator theory and has been widely studied due to unexpected intimate connections with differential equations, particularly conjugate point theory and disconjugacy. For positive integers m and k , an operator T is said to be a k -quasi-m-symmetric operator if T ∗k ( ∑m j=0 (−1)j ( m j )T ∗m−jT j )T k = 0, which is a generalization of m-symmetric operator. In this paper, some basic structural properties of k -quasi-m-symmetric operators are established with the help of operator matrix representation. In particular, we also show that every k -quasi-3-symmetric operator has a scalar extension. Finally, we prove that generalized Weyl’s theorem holds for k -quasi-3-symmetric operators.