Finite rank perturbations of complex symmetric operators

$C$-normality of rank-one perturbations of normal operators

“…Let N ∈ B(H) be a normal operator. In [6], the authors proved that if U ∈ B(H) is unitary and limit (in strong operator topology) of operators of the form P(N, N * ) with P ∈ C[X, Y], then N + λUx ⊗ x ∈ S(H) for all x ∈ H and λ ∈ C. Later in [1], this result is shown to remain valid to all unitary operators commuting with N. Since GS(H) contains all complex symmetric operators, then one may expect that a larger class of rank-one perturbations of normal operators must lie in GS(H). This is indeed the case.…”