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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Invariance of projections in the diagonal of a nest algebra


Author: John Daughtry
Journal: Proc. Amer. Math. Soc. 102 (1988), 117-120
MSC: Primary 47C05,; Secondary 47D25
MathSciNet review: 915727
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Abstract: The study of operator factorization along commutative subspace lattices which are not nests leads to the investigation of the mapping $ {\phi _A}$ which takes an orthogonal projection $ Q$ in the diagonal of a nest algebra $ \mathcal{A}$ to the projection on the closure of the range of AQ for certain bounded linear operators $ A$. The purpose of this paper is to demonstrate that if $ B$ is an operator leaving the range of $ Q$ invariant, $ V$ is an element of the "Larson radical" of $ \mathcal{A},B + V$ is invertible, $ {(B + V)^{ - 1}}$ belongs to $ \mathcal{A}$, and $ {\phi _{B + V}}(Q)$ is in the diagonal of $ \mathcal{A}$, then $ {\phi _V}(Q) \leq Q$. For example, if $ V$ is in the Jacobson radical of $ \mathcal{A}$ and $ \lambda $ is a nonzero scalar, it follows that $ {\phi _{\lambda I + V}}(Q) = Q$ if and only if $ {\phi _{\lambda I + V}}(Q)$ belongs to the diagonal of $ \mathcal{A}$. Examples of the applications to operator factorization and unitary equivalence of sets of projections are given.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1988-0915727-7
PII: S 0002-9939(1988)0915727-7
Keywords: Nest algebra, CSL algebra, factorization of positive operators
Article copyright: © Copyright 1988 American Mathematical Society