Adjoint of sums and products of operators in Hilbert spaces

Abstract

We provide sufficient and necessary conditions guaranteeing equations $(A+B{)}^{\ast }=A^{\ast }+B^{\ast }$ and $(AB{)}^{\ast }=B^{\ast }{A}^{\ast }$ 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 $M_{S,T}=\left(\begin{array}{cc}I& -T\\ S& I\end{array}\right)$ that makes it possible to treat real and complex Hilbert spaces jointly.

arXiv comment13 pages