New double inequalities for g-frames in Hilbert [Formula: see text]-modules.

Zhong-Qi Xiang

doi:10.1186/s40064-016-2731-2

Abstract

In this work, we establish two types of double inequalities for g-frames in Hilbert C^{*}-modules, which involve scalars lambda in [0,1] and lambda in [frac{1}{2},1] respectively. It is shown that the results we obtained can immediately lead to the existing corresponding results when taking lambda =frac{1}{2}.

Highlights

The origins of the notion of frames can be traced back to the literature Duffin and Schaeffer (1952) in the early 1950’s, when they were used to deal with some problems in nonharmonic Fourier series
The concepts of frames and g-frames for Hilbert spaces have been generalized to the case of Hilbert C∗-modules (Frank and Larson 2002; Khosravi and Khosravi 2008)
It should be pointed out, due to the complex structure of C∗-algebras embedded in the Hilbert C∗-modules, that the problems about frames and g-frames in Hilbert C∗-modules are more complicated than those in Hilbert spaces

Summary

Introduction

The origins of the notion of frames can be traced back to the literature Duffin and Schaeffer (1952) in the early 1950’s, when they were used to deal with some problems in nonharmonic Fourier series. Let { j ∈ End∗A(H, Kj)}j∈J be a g-frame for H with respect to {Kj}j∈J, the g-frame operator S for { j}j∈J is defined by The authors of Xiao and Zeng (2010) have already extended the inequalities for Parseval frames and general frames to g-frames in Hilbert C∗-modules: Theorem 1 Let { j ∈ End∗A(H, Kj)}j∈J be a g-frame for H with respect to {Kj}j∈J, and { j}j∈J be the canonical dual g-frame of { j}j∈J, for any K ⊂ J and any f ∈ H , we have

Results

Conclusion