Classification of General Sequences by Frame-Related Operators

TL;DR

This paper surveys and extends the analysis of operators associated with sequences, including Bessel sequences and frames, exploring their properties and classifications, especially for arbitrary sequences and their relation to orthonormal bases.

Contribution

It introduces new characterizations of sequences via operators, including unbounded cases, and classifies sequences based on operators applied to orthonormal bases.

Findings

Operators for arbitrary sequences can be unbounded.

Sequences are characterized by properties of associated operators.

Sequences can be classified through operators on orthonormal bases.

Abstract

This note is a survey and collection of results, as well as presenting some original research. For Bessel sequences and frames, the analysis, synthesis and frame operators as well as the Gram matrix are well-known, bounded operators. We investigate these operators for arbitrary sequences, which in general lead to possibly unbounded operators. We characterize various classes of sequences in terms of these operators and vice-versa. Finally, we classify these sequences by operators applied on orthonormal bases.