Is $B - B'$ self-adjoint provided $B,B'$ are positive operators? If I have two positive operators $B,B'$ on an arbitrary Hilbert space $\mathcal{H}$ not necessarily over $\mathbb{C}$, how do I know that $B - B'$ is self adjoint? EDIT: Reed and Simon define positiveness of an operator $B$ as follows: We say that an operator $B \colon \mathcal{H} \to \mathcal{H}$ is positive if for every $x \in \mathcal{H}$, $ \langle x,Bx \rangle \ge 0 \,. $

As Daniel Fischer said: if this is how positivity is defined, and if the ground field is $\mathbb R$, then the statement is false. Just let $B'$ be the zero operator, and let $B$ be any non-self-adjoint positive operator such as $$\begin{pmatrix} 2 & 1 \\\ 0 & 2 \end{pmatrix}$$ The authors either implicitly switched to the more common definition of positivity (which requires self-adjointness), or omitted a hypothesis.