Large Perturbations of Nest Algebras

TL;DR

This paper investigates the stability of nest algebras under perturbations, showing that small changes in nests lead to similar algebras, but also identifying cases where small perturbations do not preserve algebra proximity.

Contribution

It establishes bounds relating nest distances to algebra similarities and provides examples where small nest perturbations do not result in small algebra perturbations.

Findings

Nest algebras are close if nests are close within a distance less than 1.

Similar nest algebras can be obtained via invertible transformations when nests are within distance less than 1.

Counterexamples show nests closer than 1 can have their algebras at distance 1.

Abstract

Let $\mathcal{M}$ and $\mathcal{N}$ be nests on separable Hilbert space. If the two nest algebras are distance less than 1 ($d(\mathcal{T}(\mathcal{M}),\mathcal{T}(\mathcal{N})) < 1$), then the nests are distance less than 1 ($d(\mathcal{M},\mathcal{N})<1$). If the nests are distance less than 1 apart, then the nest algebras are similar, i.e. there is an invertible $S$ such that $S\mathcal{M} = \mathcal{N}$, so that $S \mathcal{T}(\mathcal{M})S^{-1} = \mathcal{T}(\mathcal{N})$. However there are examples of nests closer than 1 for which the nest algebras are distance 1 apart.