Nests with the partial factorization property

It is proved that a nest $\mathcal N$ on a separable complex Hilbert space $\mathcal H$ has the left (resp. right) partial factorization property, which means that for every invertible operator $T$ from $\mathcal H$ onto a Hilbert space $\mathcal K$ there exists an isometry (resp. a coisometry) $U$ from $\mathcal H$ into $\mathcal K$ such that both $U^*T$ and $T^{-1}U$ are in the associated nest algebra $Alg \mathcal N$ if and only if it is atomic (resp. countable).