Factorisation in nest algebras. II

Abstract

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