Loops in SU(2) and Factorization, II

TL;DR

This paper extends previous results on SU(2) loops with critical smoothness to loops of vanishing mean oscillation, linking Toeplitz operator invertibility with unique factorizations, using operator-theoretic methods.

Contribution

It generalizes the equivalence between Toeplitz invertibility and loop factorizations from critical smoothness to vanishing mean oscillation loops, via operator-theoretic approaches.

Findings

Generalization to VMO loops achieved

Operator-theoretic factorization developed

Implications for loop factorization theory discussed

Abstract

In the prequel to this paper, we proved that for a $SU(2,\mathbb C)$ valued loop having the critical degree of smoothness (one half of a derivative in the $L^2$ Sobolev sense), the following statements are equivalent: (1) the Toeplitz and shifted Toeplitz operators associated to the loop are invertible, (2) the loop has a unique triangular factorization, and (3) the loop has a unique root subgroup factorization. This hinges on some Plancherel-esque formulas for determinants of Toeplitz operators. The main point of this report is is to outline a generalization of this result to loops of vanishing mean oscillation, and to discuss some consequences. This generalization hinges on an operator-theoretic factorization of the Toeplitz operators (not simply their determinants).