On (A,m)-Symmetric Operators in a Hilbert Space

For a positive AB(H), an operator TB(H) is said to be (A,m)-symmetric if it satisfies the operator equation.Y This class of operators seems a natural generalization of m-symmetric operators on a Hilbert space. In this paper, first we give various properties related to such a family. Then, we prove that if T and Q are commuting operators, T is (A,m)-symmetric and Q is l-nilpotent, then (T+Q) is (A,m+2l-2)-symmetric. In addition, we show that every power of an (A,m)-symmetric operator is also (A,m)-symmetric. Some connection between (A,m)-symmetric operators and C0-semigroups are also shown. Finally, we characterize the spectra of such operators.