Hereditarily and nonhereditarily complete systems of vectors in a Hilbert space

In this paper, we study the property of hereditary completeness of vector systems $\{x_k{\}}_{k=1}^{\mathrm{\infty }}$ in a Hilbert space. A criterion of hereditary completeness is obtained in terms of projectors on closed linear spans of systems of the form $\{x_k{\}}_{k\in N}$, $N\subset \mathbb{N}$. Developed technique has been used to prove that mixed systems of a hereditarily complete system are also hereditarily complete. In conclusion, the problem of possible defects in a nonhereditarily complete system is considered.