## Pomona Faculty Publications and Research

#### Document Type

Article - preprint

#### Department

Mathematics (Pomona)

2012

#### Keywords

Complex symmetric operator, Unitary orbit, Compact operator, Strong operator topology, Weak operator topology, Kakutani shift, Self-similarity, Palindrome, Shift operator, Unilateral shift

#### 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.

Pre-print from http://arxiv.org/abs/1106.4855

Final publication can be found at:

Stephan Ramon Garcia, Daniel E. Poore, On the closure of the complex symmetric operators: Compact operators and weighted shifts, Journal of Functional Analysis, Volume 264, Issue 3, 1 February 2013, Pages 691-712, ISSN 0022-1236, http://dx.doi.org/10.1016/j.jfa.2012.11.009. (http://www.sciencedirect.com/science/article/pii/S0022123612004181)