Czechoslovak Mathematical Journal, Vol. 72, No. 1, pp. 239-257, 2022
Compact operators and integral equations in the $\mathcal{HK}$ space
Varayu Boonpogkrong
Received October 15, 2020. Published online July 29, 2021.
Abstract: The space $\mathcal{HK}$ of Henstock-Kurzweil integrable functions on $[a,b]$ is the uncountable union of Fréchet spaces $\mathcal{HK}(X)$. In this paper, on each Fréchet space $\mathcal{HK}(X)$, an $F$-norm is defined for a continuous linear operator. Hence, many important results in functional analysis, like the Banach-Steinhaus theorem, the open mapping theorem and the closed graph theorem, hold for the $\mathcal{HK}(X)$ space. It is known that every control-convergent sequence in the $\mathcal{HK}$ space always belongs to a $\mathcal{HK}(X)$ space for some $X$. We illustrate how to apply results for Fréchet spaces $\mathcal{HK}(X)$ to control-convergent sequences in the $\mathcal{HK}$ space. Examples of compact linear operators are given. Existence of solutions to linear and Hammerstein integral equations is proved.
Keywords: compact operator; integral equation; controlled convergence; Henstock-Kurzweil integral
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Affiliations: Varayu Boonpogkrong, Department of Mathematics, Division of Computational Science, Faculty of Science, Prince of Songkla University, Hat Yai, 90110 Thailand, e-mail: varayu.b@psu.ac.th
