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

TL;DR

This paper investigates the closure of complex symmetric operators on infinite-dimensional Hilbert spaces, showing that all compact operators in this closure are complex symmetric and exploring properties of weighted shifts in the closure but outside the set of symmetric operators.

Contribution

It proves that every compact operator in the closure of complex symmetric operators is itself complex symmetric and characterizes certain weighted shifts in this closure.

Findings

All compact operators in the closure are complex symmetric.

Weighted shifts with approximate self-similarity are in the closure but not symmetric.

The results on compact operators are shown to be optimal.

Abstract

We study the closure $\bar{CSO}$ 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 $\bar{CSO}$ is complex symmetric. Using a construction of Kakutani as motivation, we also describe many properties of weighted shifts in $\bar{CSO} \backslash CSO$. In particular, we show that weighted shifts which demonstrate a type of approximate self-similarity belong to $\bar{CSO}\backslash CSO$. As a byproduct of our treatment of weighted shifts, we explain several ways in which our result on compact operators is optimal.