Invariance of projections in the diagonal of a nest algebra

Abstract

The study of operator factorization along commutative subspace lattices which are not nests leads to the investigation of the mapping ϕ A {\phi _A} which takes an orthogonal projection Q Q in the diagonal of a nest algebra A \mathcal {A} to the projection on the closure of the range of AQ for certain bounded linear operators A A . The purpose of this paper is to demonstrate that if B B is an operator leaving the range of Q Q invariant, V V is an element of the "Larson radical" of A , B + V \mathcal {A},B + V is invertible, ( B + V ) − 1 {(B + V)^{ - 1}} belongs to A \mathcal {A} , and ϕ B + V ( Q ) {\phi _{B + V}}(Q) is in the diagonal of A \mathcal {A} , then ϕ V ( Q ) ≤ Q {\phi _V}(Q) \leq Q . For example, if V V is in the Jacobson radical of A \mathcal {A} and λ \lambda is a nonzero scalar, it follows that ϕ λ I + V ( Q ) = Q {\phi _{\lambda I + V}}(Q) = Q if and only if ϕ λ I + V ( Q ) {\phi _{\lambda I + V}}(Q) belongs to the diagonal of A \mathcal {A} . Examples of the applications to operator factorization and unitary equivalence of sets of projections are given.