O-frames for operators in Banach spaces

TL;DR

The paper introduces a new concept of O-frames for operators in Banach spaces, establishing their equivalence with the bounded approximation property and factorization through spaces with bases, with applications provided.

Contribution

It presents a novel notion of O-frames for operators in Banach spaces and characterizes operators with O-frames via the BAP and factorization properties.

Findings

Operator has an O-frame iff it has the BAP

Operators with O-frames can be factored through spaces with bases

Applications demonstrate the usefulness of O-frames in analysis

Abstract

These notes are formal. Here, in this abstract, not in the note, we should say that all that is in the text was done, essentially, by Aleksander Pe{\l}czy\'nski. BUT: Anyhow, a new notion of an O-frame for an operator is introduced. For the operators in separable spaces, it is shown that a operator has an O-frame iff it has the BAP iff it can be factored through a Banach space with a basis. Applications are given. However, looking around, I'd say that, e.g., a notion of a Banach frame (and also O-frame) was implicitely introduced by great Aleksander Pe{\l}czy\'nski.