A New Representation and Characterization of Riesz Bases

Normally, all Riesz bases in a Hilbert space $$\mathcal{H}$$ are characterized in terms of invertible bounded linear operators acting on a single orthonormal basis. In this paper, it is demonstrated that a Riesz basis can be represented by a narrower class of linear operators on a unique orthonormal basis. This new representation defines an equivalence relation on the set of all Riesz bases and characterizes Riesz bases using the concept of biframes. Also, some results on the topic of biframes are obtained using this characterization.