Multipliers for Lipschitz p-Bessel sequences in metric spaces

TL;DR

This paper extends the concept of multipliers to Lipschitz functions in metric spaces, introduces Lipschitz frames, and explores their properties and effects on operator compactness and parameter variation.

Contribution

It introduces Lipschitz multipliers and frames in metric spaces, expanding the framework beyond Banach spaces and analyzing their operator properties.

Findings

Lipschitz multipliers can be compact when the symbol sequence converges to zero.

Lipschitz frames generalize Bessel sequences to metric spaces.

Parameter variations influence the properties of multipliers.

Abstract

The notion of multipliers in Hilbert space was introduced by Schatten in 1960 using orthonormal sequences and was generalized by Balazs in 2007 using Bessel sequences. This was extended to Banach spaces by Rahimi and Balazs in 2010 using p-Bessel sequences. In this paper, we further extend this by considering Lipschitz functions. On the way we define frames for metric spaces which extends the notion of frames and Bessel sequences for Banach spaces. We show that when the symbol sequence converges to zero, the multiplier is a Lipschitz compact operator. We study how the variation of parameters in the multiplier effects the properties of multiplier.