Let $A$ be a bounded self-adjoint operator on a Hilbert space. Reed and Simon I, page 223, it says that for $P(A) = \sum_{n=0}^\infty a_n A^n$ we have
$$ \|P(A)\|^2 = \|P(A)^*P(A)\| = \|(\overline P P)(A)\| $$
Why do these two equalities hold? I'm not even sure I understand what is meant by the notation '$(\overline P P)(A)$'?
If $P$ is a power series $P(x) = \sum_{n=0}^\infty a_nx^n$ then $\overline{P}(x) = \sum_{n=0}^\infty \overline{a_n}x^n$ is the power series with complex conjugation applied to every coefficient.
The identity $\|T\|^2 = \|T^*T\|$ holds for any bounded map $T$, and is called the $C^*$-identity.
Furthermore, $T \mapsto T^*$ is antilinear, continous and antimultiplicative so:
$$P(A)^* = \left(\sum_{n=0}^\infty a_nA^n\right)^* = \sum_{n=0}^\infty \overline{a_n}(A^*)^n = \sum_{n=0}^\infty \overline{a_n}A^n = \overline{P}(A)$$
If $P(x) = \sum_{n=0}^\infty a_nx^n$ and $Q(x) = \sum_{n=0}^\infty b_nx^n$ are two power series, then $$(PQ)(x) = \sum_{n=0}^\infty \left(\sum_{k=0}^n a_nb_{n-k}\right)x^n$$ is the product of $P$ and $Q$.
The equality $P(T)Q(T) = (PQ)(T)$ clearly holds for any bounded map $T$.
Putting this together:
$$\left\|P(A)\right\|^2 = \left\|P(A)^*P(A)\right\| = \left\|(\overline P P)(A)\right\|$$