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

Abstract

Let $X$ be a completely regular Hausdorff space and $Cb(X)$ 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 $T\\colon Cb(X) \\to E$ 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.