Cross-frames in Banach spaces

TL;DR

This paper extends the concept of dual frames from Hilbert spaces to Banach spaces, analyzing properties of synthesis operators and showing limitations on certain operator groups related to cross-frames.

Contribution

It introduces the generalization of dual frames to Banach spaces and investigates associated synthesis operators and their limitations.

Findings

Cross-frames in Banach spaces cannot be eigenvectors of strongly continuous bounded operator groups.

Properties of synthesis operators for dual frames in Banach spaces are characterized.

Main result shows a fundamental limitation in the structure of cross-frames in Banach spaces.

Abstract

This paper generalizes results for alternate dual frames in Hilbert spaces on the situation of a Banach space. Additionally some properties of synthesys operator associated with alternate dual frame are investigated. The main result is that for a given cross-frame in Banach space we can not find such strongly continuous uniformely bounded one-parameter group of operators that the vectors from this cross-frame are eigenvectors of group's operators.