Adjoint of sums and products of operators in Hilbert spaces

Abstract: 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 W\"ust. Our method involves the range of two-by-two matrices of the form $M_{S,T}=\left(\begin{array}{cc}!! I & -T!!\ !! S& I!!\end{array}\right)$ that makes it possible to treat real and complex Hilbert spaces jointly.