Adjoint of Sums and Products of Operators in Hilbert Spaces

We provide sufficient and necessary conditions guaranteeing equations (A + B)* = A* + B* and (AB)* = B*A* concerning densely defined unbounded operators A, B between Hilbert spaces. We also improve the perturbation theory of selfadjoint and essentially selfadjoint operators due to Nelson, Kato, Rellich, and Wust. Our method involves the range of two-by-two matrices of the form {IEq1} that makes it possible to treat real and complex Hilbert spaces jointly.