Some results on higher order symmetric operators

Abstract

For a positive A\in \mathcal{B}(\mathcal{H}), positive integers m and k, an operator T\in \mathcal{B}(\mathcal{H}) is called k-quasi-(A,m)-symmetric if  T^{*k}(\sum\limits_{j=0}^{m}(-1)^{j}(^{m}_{j})T^{*m-j}AT^{j})T^{k}=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.