On complete Riesz--Fischer sequences in a Hilbert space

E. Zikkos

Problemy analiza, Tome 13 (2024) no. 1, pp. 124-131

Résumé

We prove that if $\{f_n\}_{n=1}^{\infty}$ is a complete Riesz–Fischer sequence in a separable Hilbert space $H$, then $$ T:=\{f\in H\colon \sum |\langle f, f_n\rangle |^2\infty\} $$ is closed in $H$ if and only if $\{f_n\}_{n=1}^{\infty}$ has a biorthogonal Riesz sequence. If the latter is also complete in $H$, then $\{f_n\}_{n=1}^{\infty}$ is a Riesz basis for $H$.

Détail

Keywords: Riesz–Fischer sequences, Bessel sequences, Riesz sequences, Riesz bases, completeness.
Mots-clés : biorthogonal sequences