Errata to “Normalizers of nest algebras”

Abstract

In [1], Proposition 5 has an error. The proposition purports to show that SN , the normalizing semigroup for the nest algebra N , is closed in the strong operator topology. The proof begins with the net Tλ in SN converging strongly to T ∈ B(H). It then proceeds to (correctly) establish the first of the two criteria set forth in Theorem 2 needed to conclude that T ∈ SN . The last sentence of the proof then asserts that the second criterion can be established in a similar manner by just replacing all the operators with their adjoints. This is the problem: since the adjoint operation is not strongly continuous, we cannot assume that T ∗ λ converges strongly to T ∗, which is needed to make the analogous argument work to establish the second criterion. The following replacement for Proposition 5 fixes this, and strengthens the proposition. Since the adjoint operation is weakly continuous, we really do need only establish the first criterion, the second one following by replacing the operators involved with their adjoints.