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 $T\colon C_b(X) \to E$ 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 $T\colon C_b(X) \to \mathcal{A}$. As an application, we develop spectral theory for operators $T\colon C_b(X) \to \mathcal{L}_s(\mathcal{Y})$, 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.