A question about finite-rank projection on Hilbert space

Question

Let $H$ be a Hilbert space and $P_{n}\in B(H)$ be an increasing net of finite-rank projection which converge to the identity in the strong operator topology. Then,

Can we verify that $P_{n}TP_{n}\rightarrow T$ in norm topology, for any $T\in B(H)$?

Answer 1

It is not true even for $T = I$. For example if $H = \ell^2$, and $P_n$ are the projections to the first $n$ entries, then $P_n$ converges to $I$ in strong operator topology but not in norm topology.

Actually, your assertion is true only if $H$ is finite dimensional. From $P_n TP_n \to T$ in norm topology and plug in $T=I$, then $P_n$ converges to $I$ in norm topology. Thus $I$ is also a compact operator (as $P_n$ are all compact operators), which means that $H$ must be finite dimensional.