Complex symmetric operator, normal operator, binormal operator, nilpotent operator, idempotent, partial isometry
We say that an operator $T \in B(H)$ is complex symmetric if there exists a conjugate-linear, isometric involution $C:H\to H$ 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 $(\dim \ker T, \dim \ker T^*)$.
© 2010 American Mathematical Society
Garcia, S.R., Wogen, W.R., Some new classes of complex symmetric operators, Trans. Amer. Math. Soc. 362 (2010), 6065-6077. MR2661508 (2011g:47086)