Some results on higher order symmetric operators

Abstract

For some operator A ? B(H), positive integers m and k, an operator T ? B(H) is called k-quasi-(A,m)-symmetric if T*k( mP j=0 (?1)j(m j )T*m?jATj)Tk = 0, which is a generalization of the m-symmetric operator. In this paper, some basic structural properties of k-quasi-(A,m)-symmetric operators are established with the help of operator matrix representation. We also show that if T and Q are commuting operators, T is k-quasi-(A,m)-symmetric and Q is n-nilpotent, then T + Q is (k + n ? 1)-quasi-(A,m + 2n ? 2)-symmetric. In addition, we obtain that every power of k-quasi-(A,m)-symmetric is also k-quasi-(A,m)-symmetric. Finally, some spectral properties of k-quasi-(A,m)-symmetric are investigated.