Student Co-author

Pomona Undergraduate

Document Type

Article - preprint


Mathematics (Pomona)

Publication Date



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


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

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, (

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© 2013 Elsevier B.V.

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