Hereditarily and nonhereditarily complete systems of vectors in a Hilbert space

TL;DR

This paper investigates hereditary completeness of vector systems in Hilbert spaces, providing a criterion based on projectors, and explores properties of mixed systems and potential defects in nonhereditarily complete systems.

Contribution

It introduces a new criterion for hereditary completeness using projectors and analyzes the structure of mixed systems and defects in nonhereditarily complete systems.

Findings

Established a criterion for hereditary completeness in terms of projectors.

Proved that mixed systems of a hereditarily complete system are also hereditarily complete.

Discussed possible defects in nonhereditarily complete systems.

Abstract

In this paper, we study the property of hereditary completeness of vector systems $\{x_k\}_{k=1}^\infty$ 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.