# Can you have a weakly convergent sequence of unbounded linear operators? (Example)

Is it at all feasible to have a weakly convergent sequence of unbounded linear operators? If so, what is a concrete example of a sequence of not necessarily bounded linear operators who converge in the weak operator topology?

Is it in fact the case that being a sequence of operators convergent in the weak operator topology necessitates that one in fact has a sequence of bounded linear operators?

## 1 Answer

Take any unbounded operator $$T$$, and let $$T_n=\tfrac1n\,T$$. Then, for any $$x$$ in the domain of $$T$$, you have $$\|T_nx\|=\tfrac1n\|Tx\|\to0.$$ So $$T_n\to0$$ in the strong operator topology.

• I guess that this highlights a strategy for finding examples of operators convergent in the weak operator topology: look for those convergent under the strong operator topology from which it follows that they converge under the weak operator topology? – Jeremy Jeffrey James Mar 8 at 18:50
• I wouldn't call it a "strategy". You want a property for the wot, but it turns out it also occurs for the sot. That's not true in general, but it is very often far more easy to test sot convergence that it is wot convergence (because in wot convergence, as soon as you use Cauchy-Schwarz you are going into sot territory). – Martin Argerami Mar 8 at 18:57