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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.

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    Did you inspect the properties of adjoint operators? What more do you want?2011-08-24
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    In regards to the Corollary: let $(,)$ be the Hermitian inner product on $\mathbb{C}$, so $(w,z) = \bar{w}z$. Let $u$ be a complex number, consider $(uz,z) = \bar{u}\bar{z}z = \bar{u}|z|^2$, so $(uz,z)\in\mathbb{R}$ iff $u$ is real.2011-08-24
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    Note that if $\lambda$ is a complex number, then $(\lambda T)^*=\overline\lambda T^*$, and in particular the adjoint coincides with complex conjugation on scalar multiples of the identity.2011-08-24

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