Some properties of operator-valued frames

TL;DR

This paper introduces a new formula for operator-valued frames in finite-dimensional Hilbert spaces and applies it to approximate g-frames by Parseval frames and their duals with optimal estimates.

Contribution

It provides a novel formula for operator-valued frames and simplifies the approximation of g-frames by Parseval frames, including optimal dual approximations.

Findings

New formula for operator-valued frames in finite dimensions

Simplified approximation of g-frames by Parseval frames

Optimal estimates for approximation by dual frames

Abstract

Operator-valued frames (or g-frames) are generalizations of frames and fusion frames and have been used in packets encoding, quantum computing, theory of coherent states and more. In this paper, we give a new formula for operator-valued frames for finite dimensional Hilbert spaces. As an application, we derive in a simple manner a recent result of A. Najati concerning the approximation of g-frames by Parseval ones. We obtain also some results concerning the best approximation of operator-valued frames by its alternate duals, with optimal estimates.