Nests with the partial factorization property

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