Hereditarily and nonhereditarily complete systems of vectors in a Hilbert space
Mikhail Prokofyev
TL;DR
Addresses the problem of characterizing hereditary completeness for complete minimal systems in a separable Hilbert space and understanding defects in nonhereditarily complete systems. The authors develop a vector replacement lemma and a projector-based criterion, introducing subspaces $H_\sigma=\overline{Lin}\{x_k\}_{k\in\sigma}$ and $H'_\sigma=\overline{Lin}\{x_k^*\}_{k\in\sigma^c}$ with projectors $P_\sigma$ and $P'_\sigma$, and show that hereditary completeness is equivalent to the pointwise convergence $P_{\sigma_m}\to P_\sigma$ for all $\sigma$, together with a topological/compactness criterion for the projector space $\Omega(x_k)$. They prove that if a system is hereditarily complete then all of its mixed systems are hereditarily complete. They construct, for any prescribed defect set $S$ containing $0$ (and possibly $\infty$), a complete minimal system whose defects with respect to all mixed systems realize exactly $S$, providing explicit finite-case and infinite-case constructions. These results unify reformulations of hereditary completeness and offer a method to engineer vector systems with prescribed defect structures, with implications for basis theory and spectral analysis.
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.