Decomposition of a unitary operator on a non-separable Hilbert space

Question

In Halmos's "A Hilbert space problem book" we read "If $H$ is non-separable, then it is the direct sum of separable infinite-dimensional subspaces that reduce $A$ [$A$ is any normal operator]..." [Problem 142, p. 267], i.e. $H=\underset{a\in I}{\oplus}H_{a}$ where the index set I is uncountable. If $A$ is a unitary operator, does that mean that we could write it as $A=\underset{a\in I}{\sum}e^{i\phi_{\alpha}}P_{a} $ where $P_{a}$ is the projection onto $H_{a}$ and $\phi_{a}\in[0,2\pi]$ or do we need to assume further that $Α$ has a pure point spectrum?