Invertibility in nest algebras

Abstract:

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