On the relation of the frame-related operators of fusion frame systems

TL;DR

This paper explores the relationships between frame-related operators in fusion frame systems, providing operator identities and properties that enhance understanding of their structure and applications.

Contribution

It offers a detailed analysis of the operators in fusion frame systems, establishing new identities and properties that connect local and global frame operators.

Findings

Derived operator identities linking local and global frame operators

Characterized bounded block diagonal operators in Hilbert direct sums

Enhanced understanding of fusion frame system properties

Abstract

Frames have been investigated frequently over the last few decades due to their valuable properties, which are desirable for various applications as well as interesting for theory. Some applications additionally require distributed processing techniques, which naturally leads to the concept of fusion frames and fusion frame systems. The latter consists of a system of subspaces, equipped with local frames on each of them, and a global frame. In this paper, we investigate the relations of the associated frame-related operators on all those three levels. For that we provide a detailed investigation on bounded block diagonal operators between Hilbert direct sums. We give the relation of the frame-related operators of the fusion frame and the corresponding frame systems in terms of operator identities. By applying these identities we prove further properties of fusion frame systems.