The canonical form of a scalar operator on a Banach space

Abstract

Let A = ∫ λ d E ( λ ) A = \smallint \lambda dE(\lambda ) be a scalar operator on a Banach space X. If there exists a vector g ∈ X g \in X such that the closed convex hull of the range of the vector measure μ ( ⋅ ) = E ( ⋅ ) g \mu ( \cdot ) = E( \cdot )g has nonvoid interior, then A is similar to the operator Q f ( λ ) = λ f ( λ ) Qf(\lambda ) = \lambda f(\lambda ) on a quotient space of a suitably constructed L ∞ {\mathcal {L}^\infty } space.