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

2011/06/23 by Stephan Ramon Garcia, Garcia, Stephan Ramon, Daniel E. Poore +1
Mathematics · #47A05 #47B35 #47B99 #Advanced Banach Space Theory #Advanced Operator Algebra Research #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Operator Algebras (math.OA) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1106.4855

openalex publication_date 2011/06/23 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28

Abstract

We study the closure \CSO of the set CSO of all complex symmetric\noperators on a separable, infinite-dimensional, complex Hilbert space. Among\nother things, we prove that every compact operator in \CSO is complex\nsymmetric. Using a construction of Kakutani as motivation, we also describe\nmany properties of weighted shifts in \CSO backslash CSO. In\nparticular, we show that weighted shifts which demonstrate a type of\napproximate self-similarity belong to \CSO backslash CSO. As a byproduct\nof our treatment of weighted shifts, we explain several ways in which our\nresult on compact operators is optimal.\n

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