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We provide sufficient and necessary conditions guaranteeing equations$(A+B)^*=A^*+B^*$ and $(AB)^*=B^*A^*$ concerning densely defined unboundedoperators $A,B$ between Hilbert spaces. We also improve the perturbation theoryof selfadjoint and essentially selfadjoint operators due to Nelson, Kato,Rellich, and W\"ust. Our method involves the range of two-by-two matrices ofthe form $M_{S,T}=\left(\begin{array}{cc}\!\! I & -T\!\!\\ \!\! S&I\!\!\end{array}\right)$ that makes it possible to treat real and complexHilbert spaces jointly.

Adjoint of sums and products of operators in Hilbert spaces