Integration with respect to Baire vector measures with applications to the spectral theory

Abstract

Let $X$ be a completely regular Hausdorff space and $C_b(X)$ the space of all bounded continuous scalar functions on $X$, equipped with the strict topology ${\beta }_{\sigma }$. We develop a general integral representation theory of $({\beta }_{\sigma },\xi )$-continuous linear operators from $C_b(X)$ to a locally convex Hausdorff space $(E,\xi )$. We present equivalent conditions for a $({\beta }_{\sigma },\xi )$-continuous operator $T:C_b(X)\to E$ to be weakly compact. If $(\mathcal{A},\xi )$ is a sequentially complete unital locally convex algebra, we establish an integral representation of a $({\beta }_{\sigma },\xi )$-continuous unital algebra homomorphism $T:C_b(X)\to \mathcal{A}$. As an application, we develop spectral theory for operators $T:C_b(X)\to {\mathcal{L}}_s(\mathcal{Y})$, where ${\mathcal{L}}_s(\mathcal{Y})$ denotes the algebra of bounded linear operators of a Banach space $\mathcal{Y}$ into itself, equipped with the topology ${\tau }_s$ of simple convergence.