FACTORIZATION IN NEST-ALGEBRAS - II

Citation
M. Anoussis et Eg. Katsoulis, FACTORIZATION IN NEST-ALGEBRAS - II, Transactions of the American Mathematical Society, 350(1), 1998, pp. 165-183
Citations number
23
Categorie Soggetti
Mathematics,Mathematics
ISSN journal
00029947
Volume
350
Issue
1
Year of publication
1998
Pages
165 - 183
Database
ISI
SICI code
0002-9947(1998)350:1<165:FIN-I>2.0.ZU;2-7
Abstract
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)?.