Splitting of operator for frame inequalities in Hilbert spaces

TL;DR

This paper introduces a new class of inequalities for frames in Hilbert spaces based on splitting the frame operator into two positive semidefinite parts, providing a unified framework that generalizes previous results.

Contribution

It presents a novel parametrized inequality for frames and offers a transparent proof based on splitting the frame operator, unifying and extending prior results.

Findings

New parametrized inequalities for frames

Unified proof technique via operator splitting

Generalization of previous frame inequalities

Abstract

In this paper, we obtain a new type of inequalities for frames, which are parametrized by a parameter \lambda\in R . By suitable choices of {\lambda}, one obtains the previous results as special cases. Our new proof also makes the underlying mathematical structure that gives rise to these inequalities more transparent than previous approaches: Our proof shows that the main point is the splitting S = S1 + S2 of the positive denite frame operator S into the two positive semidenite operators S1 and S2 .