Let $\mathit{H}$ be a (real or complex) Hilbert space and $U:\mathit{H}\rightarrow\mathit{H}$ be a unitary operator. What conditions can be placed on $U$ to guarantee a sequence $v\_n$ such that $|v\_n|=1$ and ($Uv\_n$,$v\_n$)$\rightarrow$0 as $n\rightarrow\infty$?



Almost Orthogonal Vectors given a Unitary Operator
I'll suppose this is a complex Hilbert space and you're using the convention that the inner product is linear in the first argument and conjugate-linear in the second. The set of all possible $\langle Uv, v\rangle$ for unit vectors $v$ is the numerical range of $U$. Since unitary operators are normal, the closure of the numerical range is the convex hull of the spectrum of $U$. Thus a necessary and sufficient condition is that 0 is in the convex hull of the spectrum.


