Let $H$ be a complex Hilbert space and $(T_n)$ a monotonically nondecreasing sequence of bounded self-adjoint operators on $H$. It is well known that if there exists a bounded self-adjoint operator $K$ on $H$ such that every $T_i$ commutes with every $T_j$ and with $K$, then there exists a bounded self-adjoint operator $T$ such that $(T_n)$ strongly convergent to $T$ and $T\leq K$ (meaning $K-T$ is a positive operator).
My question is if there is an example of such a sequence of bounded self-adjoint operators (i.e. satisfying the stated conditions) but $(T_n)$ is not norm convergent to $T$? I tried google for it but it does not seem to turn anything up.
This works even without the commutativity assumptions. In any case, here is the canonical example.
Fix an increasing sequence of subspaces $H_n$ with $\bigcup_nH_n$ dense in $H$. Let $T_n$ be the orthogonal projection onto $H_n$. Then $\{T_n\}$ is increasing, it converges strongly to $I$, but $\|I-T_n\|=1$ for all $n$.