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

We study the closure CSO of the set CSO of all complex symmetric operators on a separable, infinitedimensional, complex Hilbert space. Among other things, we prove that every compact operator in CSO is complex symmetric. Using a construction of Kakutani as motivation, we also describe many properties of weighted shifts in CSO\CSO. In particular, we show that weighted shifts which demonstrate a type of approximate self-similarity belong to CSO\CSO. As a byproduct of our treatment of weighted shifts, we explain several ways in which our result on compact operators is optimal. © 2012 Elsevier Inc. All rights reserved.