### Abstract

In previous work, we proved that for a (Formula presented.)-valued loop having the critical degree of smoothness (one half of a derivative in the (Formula presented.) Sobolev sense), the following 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. For a loop (Formula presented.) satisfying these conditions, the Toeplitz determinant (Formula presented.) and shifted Toeplitz determinant (Formula presented.) factor as products in root subgroup coordinates. In this paper, we observe that, at least in broad outline, there is a relatively simple generalization to loops having values in (Formula presented.). The main novel features are that (1) root subgroup coordinates are now rational functions, i.e. there is an exceptional set and associated uniqueness issues, and (2) the noncompactness of (Formula presented.) entails that loops are no longer automatically bounded, and this (together with the exceptional set) complicates the analysis at the critical exponent.

Original language | English (US) |
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Journal | Random Matrices: Theory and Application |

DOIs | |

State | Accepted/In press - May 30 2018 |

### Keywords

- Loop groups
- root subgroup and triangular factorization
- Toeplitz operators

### ASJC Scopus subject areas

- Algebra and Number Theory
- Statistics and Probability
- Statistics, Probability and Uncertainty
- Discrete Mathematics and Combinatorics