2009/07/22 by Stephan Ramon Garcia, Garcia, Stephan Ramon, Warren R. Wogen +1 · 3 citations
Mathematics · #47B99 #Algebraic and Geometric Analysis #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.0907.3761
openalex publication_date 2009/07/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We say that an operator T ∈ B(H) is complex symmetric if there exists a conjugate-linear, isometric involution C:H→ H so that T = CT^*C. We prove that binormal operators, operators that are algebraic of degree two (including all idempotents), and large classes of rank-one perturbations of normal operators are complex symmetric. From an abstract viewpoint, these results explain why the compressed shift and Volterra integration operator are complex symmetric. Finally, we attempt to describe all complex symmetric partial isometries, obtaining the sharpest possible statement given only the data (dim ker T, dim ker T^*).