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

In this paper we prove several results on normal forms for linear displacement context-free grammars. The results themselves are rather simple and use well-known techniques, but they are extensively used in more complex constructions. Therefore this article mostly serves educational and referential purposes. 1 Displacement context-free grammars Displacement context-free grammars (DCFGs) are a r...