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.
