The set of all normal operators on a Hilbert space is not strongly closed

Question

I need an example to show that the set of all normal operators on a Hilbert space is not strongly closed. Also I know that strong operator topology and strong* operator topology coincide in the set of all normal operators, so could I conclude that this set is not strongly* closed?

Answer 1

I don't have a concrete example in my head right now.

But here is the fact: the strong limits of normal operators are precisely the subnormal operators.

Added much later: here is a path to show the above. One can show

• unitaries are wot-dense in the unit ball

• because the unit ball is convex, its sot and wot closures agree

The above then says that we can do the following: start with the unilateral shift $S$, which is subnormal. Construct a net $\{U_j\}$ of unitaries with $U_j\to S$ wot. Then construct a net $\{Q_k\}$ where each $Q_j$ is a convex combination of some $U_j$, and $Q_k\to S$ sot.