A Systematic Study of Frame Sequence Operators and their Pseudoinverses

TL;DR

This paper systematically analyzes the operators related to frame sequences in Hilbert spaces, focusing on their pseudoinverses, interactions, and classifications, especially for tight frame sequences, providing new insights into their structure.

Contribution

It offers a comprehensive study of frame sequence operators and their pseudoinverses, revealing new relationships and classifications, particularly for tight frames.

Findings

Operators are interconnected in simple ways for tight frames

Pseudoinverses of frame operators are characterized and analyzed

Classification of frame sequences based on operator properties

Abstract

In this note we investigate the operators associated with frame sequences in a Hilbert space $H$, i.e., the synthesis operator $T:\ell ^{2}(\mathbb{N}) \to H$, the analysis operator $T^{\ast}:H\to $ $% \ell ^{2}(\mathbb{N}) $ and the associated frame operator $S=TT^{\ast}$ as operators defined on (or to) the whole space rather than on subspaces. Furthermore, the projection $P$ onto the range of $T$, the projection $Q$ onto the range of $T^{\ast}$ and the Gram matrix $G=T^{\ast}T$ are investigated. For all these operators, we investigate their pseudoinverses, how they interact with each other, as well as possible classification of frame sequences with them. For a tight frame sequence, we show that some of these operators are connected in a simple way.