English

Some new classes of complex symmetric operators

Functional Analysis 2009-07-23 v1 Operator Algebras

Abstract

We say that an operator TB(H)T \in B(H) is complex symmetric if there exists a conjugate-linear, isometric involution C:HHC:H\to H so that T=CTCT = 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 (dimkerT,dimkerT)(\dim \ker T, \dim \ker T^*).

Keywords

Cite

@article{arxiv.0907.3761,
  title  = {Some new classes of complex symmetric operators},
  author = {Stephan Ramon Garcia and Warren R. Wogen},
  journal= {arXiv preprint arXiv:0907.3761},
  year   = {2009}
}

Comments

13 pages, to appear in Transactions of the AMS

R2 v1 2026-06-21T13:27:37.954Z