Bimodules of Banach space nest algebras

TL;DR

This paper extends the theory of bimodules and support functions from Hilbert space nest algebras to Banach space nest algebras, providing characterizations of bimodules and their supports.

Contribution

It develops a Banach space analogue of established Hilbert space results, characterizing bimodules via support functions and pairs, and linking reflexive bimodules to admissible support functions.

Findings

Characterization of maximal and minimal bimodules with given support functions

Weakly closed bimodules are exactly reflexive operator spaces

Unique determination of admissible support functions by reflexive bimodules

Abstract

We extend to Banach space nest algebras the theory of essential supports and support function pairs of their bimodules, thereby obtaining Banach space counterparts of long established results for Hilbert space nest algebras. Namely, given a Banach space nest algebra $\mathcal A$, we charaterise the maximal and the minimal $\mathcal A$-bimodules having a given essential support function or support function pair. These characterisations are complete except for the minimal $\mathcal A$-bimodule corresponding to a support function pair, in which case we make some headway. We also show that the weakly closed bimodules of a Banach space nest algebra are exactly those that are reflexive operator spaces. To this end, we crucially prove that reflexive bimodules determine uniquely a certain class of admissible support functions.