Adjoint of sums and products of operators in Hilbert spaces

TL;DR

This paper establishes conditions for adjoint operations on sums and products of unbounded operators in Hilbert spaces and advances the perturbation theory for selfadjoint operators using a matrix range approach.

Contribution

It provides new necessary and sufficient conditions for adjoint equations and improves existing perturbation theories with a novel matrix method.

Findings

Derived conditions for $(A+B)^*=A^*+B^*$ and $(AB)^*=B^*A^*$.

Enhanced perturbation results for selfadjoint operators.

Introduced a matrix range technique applicable to real and complex 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.