C-normality of rank-one perturbations of normal operators

Abstract

For a separable complex Hilbert space H, we say that a bounded linear operator T acting on H is C-normal, where C is a conjugation on H, if it satisfies . For a normal operator, we give geometric conditions which guarantee that its rank-one perturbation is a C-normal for some conjugation C. We also obtain some new properties revealing the structure of C-normal operators.