Characterizing Riesz Bases via Biorthogonal Riesz-Fischer sequences
Elias Zikkos
TL;DR
This paper proves that in a separable Hilbert space, biorthogonal Riesz-Fischer sequences with one complete sequence are both Riesz bases, extending previous results involving Bessel sequences.
Contribution
It establishes a new characterization of Riesz bases using biorthogonal Riesz-Fischer sequences, broadening the understanding of basis properties in Hilbert spaces.
Findings
Biorthogonal Riesz-Fischer sequences with one complete sequence are Riesz bases.
Extends previous results from Bessel sequences to Riesz-Fischer sequences.
Provides a new criterion for Riesz bases in Hilbert spaces.
Abstract
In this note we prove that if two Riesz-Fischer sequences in a separable Hilbert space are biorthogonal and one of them is complete in , then both sequences are Riesz bases for . This complements a recent result by D. T. Stoeva where the same conclusion holds if one replaces the phrase ``Riesz-Fischer sequences'' by ``Bessel sequences''.
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Taxonomy
TopicsApproximation Theory and Sequence Spaces · Advanced Banach Space Theory · Holomorphic and Operator Theory