Framings and dilations

TL;DR

This paper extends the concept of framings in Banach spaces, generalizing reconstruction formulas and significantly refining dilation results inspired by Sz-Nagy's theorem, advancing the theoretical framework of frames.

Contribution

It substantially broadens the theory of framings by extending their definitions and refining dilation results beyond existing theorems, especially those related to Sz-Nagy's dilation.

Findings

Extended the notion of framings in Banach spaces.

Refined dilation results surpassing classical Sz-Nagy theorems.

Provided new theoretical insights into the structure of frames and dilations.

Abstract

The notion of framings, recently emerging in P. G. Casazza, D. Han, and D. R. Larson, Frames for Banach spaces, in {\em The functional and harmonic analysis of wavelets and frames} (San Antonio, TX, 1999), {\em Contemp. Math}. {\bf 247} (1999), 149-182 as generalization of the reconstraction formula generated by pairs of dual frames, is in this note extended substantially. This calls on refining the basic dilation results which still being in the flavor of {\em th\'eor\`eme principal} of B. Sz-Nagy go much beyond it.