Consider $U_n$ to be a sequence of unitary operators in a separable Hilbert space $\mathcal{H}$. My questions are:
Q1) Can $U_n$ converge weakly to some projector operator $P$ as $n\to \infty$?
Q2) Can $U_n$ converge strongly to some projector operator $P$ as $n\to \infty$?
I think the answer is "yes" to both questions, if so, is there some simple example?
EDIT: $P$ is assumed to be different from the identity.
Best Answer
For 1, take $\mathcal H=\ell^2(\mathbb Z)\oplus\mathbb C$, and define $U_n$ by $U_n=U^n\oplus 1$, where $U\in B(\ell^2(\mathbb Z))$ is the unilateral shift. Then $(U_n)$ converges weakly to $0\oplus 1$, which is a rank-$1$ projection.
For 2, this cannot happen. For if $P$ is a non-trivial projection, there exists a non-zero $x\in\ker P$. Thus $\|Px\|=0$, while $\|Ux\|=\|x\|$ for all unitaries $U$.