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 ${\phi _A}$ 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.