Spectral measure of a unitary operator The following is an Exercise of Conway's operator theory:
1-Show that if $A$ is hermitian operator, then $U=\text{exp}({iA})$ is unitary.
2- Show that every unitary can be so written.
3-Find the spectral measure of the unitary in terms of that of the hermitian operator.
proof: 1-The first claim is clear.
2- I think the second is false, based on Theorem 2.1.12 of Murphy's operator theory:

And also The following from Takesaki's Operator theory:

3- There is a unique spectral measure $E$ correspondence to $*-$ homomorphism $C(\sigma(A)) \to B(H)$ and $U =\int e^{i\lambda} dE(\lambda)$.
Is it correct? Thanks.

I think that the third claim is correct with the spectral representation
$$A=\int_{\sigma(A)} \lambda \ \mathrm{d}E(\lambda)$$
