Closed operators in semi-Hilbertian spaces

Given a positive operator A on a Hilbert space, we introduce the notion of an A-closed linear operator as a natural extension of the usual notion of an A-bounded operator. We summarize a number of results and examples about this class of operators. We show that all A-bounded operators are A-closed and we prove that the class of A-closed operators is stable under perturbation by A-bounded operators. Moreover, we give sufficient conditions for the adjoint operator T* to be A-closed when T is A-closed. This study is motivated by recent developments of pseudo-Hermitian quantum mechanics.