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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.

Closed operators in semi-Hilbertian spaces