The main result of this paper is Theorem 5, which provides a necessary
and sufficient condition on a positive operator A for the existence o
f 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 Pow
er concerning outer factorisation of operators. We also show that a po
sitive operator A has the property that there exists for every nest N
an operator BN in AlgN satisfying A = BNBN (resp. A = B-N*B-N) if and
only if A is a Fredholm operator. In Section 4 we show that for a giv
en 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 so
me 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 particula
r we discuss the following question: Assume A and B are positive opera
tors such that A equal to or less than B and A belongs to F-r(N). Whic
h further conditions permit us to conclude that B belongs to F-r(N)?.