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Some facts from descriptive set theory concerning essential spectra and applications

Khalid Latrach, J. Martin Paoli, Pierre Simonnet (2005)

Studia Mathematica

Let X be a separable Banach space and denote by 𝓛(X) (resp. 𝒦(ℂ)) the set of all bounded linear operators on X (resp. the set of all compact subsets of ℂ). We show that the maps from 𝓛(X) into 𝒦(ℂ) which assign to each element of 𝓛(X) its spectrum, approximate point spectrum, essential spectrum, Weyl essential spectrum, Browder essential spectrum, respectively, are Borel maps, where 𝓛(X) (resp. 𝒦(ℂ)) is endowed with the strong operator topology (resp. Hausdorff topology). This enables us...

Some fixed point theorems and existence of weak solutions of Volterra integral equation under Henstock-Kurzweil-Pettis integrability

Afif Ben Amar (2011)

Commentationes Mathematicae Universitatis Carolinae

In this paper we examine the set of weakly continuous solutions for a Volterra integral equation in Henstock-Kurzweil-Pettis integrability settings. Our result extends those obtained in several kinds of integrability settings. Besides, we prove some new fixed point theorems for function spaces relative to the weak topology which are basic in our considerations and comprise the theory of differential and integral equations in Banach spaces.

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