Invertibility in nest algebras

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