On the closure of the complex symmetric operators: compact operators and weighted shifts

We study the closure $\overline{CSO}$ of the set $CSO$ of all complex symmetric operators on a separable, infinite-dimensional, complex Hilbert space. Among other things, we prove that every compact operator in $\overline{CSO}$ is complex symmetric. Using a construction of Kakutani as motivation, we also describe many properties of weighted shifts in $\overline{CSO}\mathrm{\backslash }CSO$. In particular, we show that weighted shifts which demonstrate a type of approximate self-similarity belong to $\overline{CSO}\mathrm{\backslash }CSO$. As a byproduct of our treatment of weighted shifts, we explain several ways in which our result on compact operators is optimal.