Representation of Operators Using Fusion Frames

TL;DR

This paper extends matrix representation methods to operator equations using fusion frames in Hilbert spaces, providing structural insights, examples, and applications to Schatten p-class operators and tensor products.

Contribution

It introduces a novel approach to representing operators with fusion frames, including structural results, isomorphism conditions, and applications to Schatten classes and tensor frames.

Findings

Fusion frame-based operator representations are structurally characterized.

Riesz decompositions yield isomorphisms in operator representations.

Tensor products of fusion frames form frames in Hilbert-Schmidt operator space.

Abstract

For finding the numerical solution of operator equations in many applications a decomposition in subspaces is needed. Therefore, it is necessary to extend the known method of matrix representation to the utilization of fusion frames. In this paper we investigate this representation of operators on a Hilbert space $\Hil$ with Bessel fusion sequences, fusion frame and Riesz decompositions. We will give the basic definitions. We will show some structural results and give some examples. Furthermore, in the case of Riesz decompositions, we prove that those functions are isomorphisms. Also, we want to find the pseudo-inverse and the inverse (if there exists) of such matrix representations. We are going to apply this idea to the Schatten $p$-class operators. Finally, we show that tensors of fusion frame are frames in the space of Hilbert-Schmidt operators.