Spectral measure of a unitary operator

Question

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.

Answer 1

The second claim is indeed true. Take $H = -i\; \log(U)$, where $\log$ is any branch of the natural logarithm that is a bounded Borel function on the unit circle.

Answer 2

I think that the third claim is correct with the spectral representation

$$A=\int_{\sigma(A)} \lambda \ \mathrm{d}E(\lambda)$$