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

Abstract

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 $\overline{CSO}\backslash CSO$. In particular, we show that weighted shifts which demonstrate a type of approximate self-similarity belong to $\overline{CSO}\backslash CSO$. As a byproduct of our treatment of weighted shifts, we explain several ways in which our result on compact operators is optimal.

19 pages, to appear in J. Funct. Anal

References in corpus (2)

• Complex symmetric partial isometries
• On a J-polar decomposition of a bounded operator and matrix representations of J-symmetric, J-skew-symmetric operators