I have been told that the adjoint of an operator behaves much like complex conjugation, and so self-adjoint operators are like real numbers. Can someone explain this analogy more in depth, or give a reference?
For example, a corollary on p. 129 of Axler's "Linear Algebra Done Right" says that the proposition
$T$ is self-adjoint iff $(Tv, v) \in \mathbb{R}$ for every $v \in V$.
is an example of how self-adjoint operators behave like real numbers. I don't see what he's talking about here, and am wondering if there are other propositions that illustrate parallels between self-adjoint operators and real numbers.