Factorisation in nest algebras

We give a necessary and sufficient condition on an operator $A$ for the existence of an operator $B$ in the nest algebra $\operatorname {Alg} \,N$ of a continuous nest $N$ satisfying $AA^*=BB^*$ (resp. $A^*A=B^*B)$. We also characterise the operators $A$ in $B(H)$ which have the following property: For every continuous nest $N$ there exists an operator $B_N$ in $\operatorname {Alg} \,N$ satisfying $AA^*=B_NB_N^*$ (resp. $A^*A=B_N^*B_N)$.