An algebraic approach of non-self-adjoint Hamiltonians in Krein spaces

Through our series of studies, we have constructed some physical operators such as non-self-adjoint Hamiltonians H, lowering operators A, and raising operators B and their adjoint H†, A†, and B† from generalized Riesz systems. However, we cannot consider the *-algebraic structure of their operators because even the sum H + H† is not well-defined. Our purpose of this paper is to introduce the *-algebra structure of all their operators by defining a certain Krein space.