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

  • Marian Nowak

    University of Zielona Gora, Poland
Integration with respect to Baire vector measures with applications to the spectral theory cover
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Abstract

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

Cite this article

Marian Nowak, Integration with respect to Baire vector measures with applications to the spectral theory. Z. Anal. Anwend. 41 (2022), no. 1/2, pp. 11–35

DOI 10.4171/ZAA/1696