Construct a unitary operator U on H with prescribed spectrum

Question

Given an infinite dimensional Hilbert space $H$. Let $|\lambda_k| = 1$ for $k = 1, ..., n$. Construct a unitary operator $U$ on $H$ such that $\sigma(U) = \{\lambda_k\}$ for $k=1,....,n.$

I can approach this question with orthonormal basis and Fourier's series.