Cut norm discontinuity of triangular truncation of graphons
Abstract
The space of $L^p$ graphons, symmetric measurable functions $w: [0,1]^2 \to \mathbb{R}$ with finite $p$norm, features heavily in the study of sparse graph limit theory. We show that the triangular cut operator $M_{\chi}$ acting on this space is not continuous with respect to the cut norm. This is achieved by showing that as $n\to \infty$, the norm of the triangular truncation operator $\mathcal{T}_n$ on symmetric matrices equipped with the cut norm grows to infinity as well. Due to the density of symmetric matrices in the space of $L^p$ graphons, the norm growth of $\mathcal{T}_n$ generalizes to the unboundedness of $M_{\chi}$. We also show that the norm of $\mathcal{T}_n$ grows to infinity on symmetric matrices equipped with the operator norm.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.06984
 Bibcode:
 2021arXiv211006984M
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Operator Algebras;
 46B28 (Primary);
 47B47 (Secondary)
 EPrint:
 10 pages, 3 figures