An unbounded operator theory approach to lower frame and Riesz-Fischer sequences

TL;DR

This paper extends unbounded operator theory to analyze lower frame and Riesz-Fischer sequences, providing new classifications, properties of duals, and generalizations of known frame results.

Contribution

It introduces a novel approach using unbounded operator theory to classify and analyze lower frame and Riesz-Fischer sequences, filling gaps in existing literature.

Findings

Classified sequences by their synthesis operator.

Extended properties of canonical duals for lower frame sequences.

Generalized results from frames to lower frame sequences.

Abstract

Frames and orthonormal bases are naturally linked to bounded operators. To tackle unbounded operators those sequences might not be well suited. This has already been noted by von Neumann in the 1920ies. But modern frame theory also investigates other sequences, including those that are not naturally linked to bounded operators. The focus of this manuscript will be two such kind of sequences: lower frame and Riesz-Fischer sequences. We will discuss the inter-relation of those sequences. We will fill a hole existing in the literature regarding the classification of those sequences by their synthesis operator. We will use the idea of generalized frame operator and Gram matrix and extend it. We will use that to show properties for canonical duals for lower frame sequences, like e.g. a minimality condition regarding its coefficients. We will also show that other results that are known for frames can be generalized to lower frame sequences. To be able to tackle these tasks, we had to revisit the concept of invertibility (in particular for non-closed operators). In addition, we are able to define a particular adjoint, which is uniquely defined for any operator.