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 Wüst. Our method involves the range of two-by-two matrices of the form $M_{S,T}=\soperator{-T}{S}$ that makes it possible to treat real and complex Hilbert spaces jointly.

DOI: 10.14232/actasm-015-809-3

AMS Subject Classification (1991): 47A05, 47A55, 47B25

Keyword(s): closed operator, adjoint, selfadjoint operator, operator product, operator sum, perturbation

Received June 10, 2014, and in revised form January 6, 2015. (Registered under 59/2015.)