The canonical form of a scalar operator on a Banach space

Abstract:

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