Invertibility in nest algebras

Abstract

Let F \mathcal {F} denote a complete nest of subspaces of a complex Hilbert space H \mathcal {H} , and let C \mathcal {C} denote the nest algebra defined by F \mathcal {F} . Let K \mathcal {K} denote the ideal of compact operators on H \mathcal {H} . If F \mathcal {F} has no infinite-dimensional gaps then T ∈ C T \in \mathcal {C} is invertible in C \mathcal {C} if and only if it is invertible in C + K \mathcal {C} + \mathcal {K} . An example is given of a nest with an infinite gap for which there exists an operator in C \mathcal {C} which is invertible in C + K \mathcal {C} + \mathcal {K} but not in C \mathcal {C} .