Factorisation in nest algebras. II

The main result of this paper is Theorem 5, which provides a necessary and sufficient condition on a positive operator $A$ for the existence of an operator $B$ in the nest algebra $AlgN$ of a nest $N$ satisfying $A=BB^{*}$ (resp. $A=B^{*}B)$. In Section 3 we give a new proof of a result of Power concerning outer factorisation of operators. We also show that a positive operator $A$ has the property that there exists for every nest $N$ an operator $B_N$ in $AlgN$ satisfying $A=B_NB_N^{*}$ (resp. $A=B_N^{*}B_N$) if and only if $A$ is a Fredholm operator. In Section 4 we show that for a given operator $A$ in $B(H)$ there exists an operator $B$ in $AlgN$ satisfying $AA^{*}=BB^{*}$ if and only if the range $r(A)$ of $A$ is equal to the range of some operator in $AlgN$. We also determine the algebraic structure of the set of ranges of operators in $AlgN$. Let $F_r(N)$ be the set of positive operators $A$ for which there exists an operator $B$ in $AlgN$ satisfying $A=BB^{*}$. In Section 5 we obtain information about this set. In particular we discuss the following question: Assume $A$ and $B$ are positive operators such that $A\leq B$ and $A$ belongs to $F_r(N)$. Which further conditions permit us to conclude that $B$ belongs to $F_r(N)$?