New Characterizations of Fusion Bases and Riesz Fusion Bases in Hilbert Spaces

TL;DR

This paper introduces new concepts and characterizations of fusion bases and Riesz fusion bases in Hilbert spaces, extending frame theory and analyzing stability and duality properties.

Contribution

It develops a novel fusion bases theory in Hilbert spaces, including dual sequences, biorthogonal sequences, and stability analysis, expanding the mathematical framework of fusion frames.

Findings

Fusion dual sequences are continuous projections.

Characterizations of Bessel and Hilbert fusion bases.

Stability of fusion bases under small perturbations.

Abstract

In this paper we investigate a new notion of bases in Hilbert spaces and similar to fusion frame theory we introduce fusion bases theory in Hilbert spaces. We also introduce a new definition of fusion dual sequence associated with a fusion basis and show that the operators of a fusion dual sequence are continuous projections. Next we define the fusion biorthogonal sequence, Bessel fusion basis, Hilbert fusion basis and obtain some characterizations of them. we study orthonormal fusion systems and Riesz fusion bases for Hilbert spaces. we consider the stability of fusion bases under small perturbations. We also generalized a result of Paley-Wiener [13] to the situation of fusion basis.