Partially integral operators with bounded kernels

Yu. Kh. Eshkabilov

Matematičeskie trudy, Tome 11 (2008) no. 1, pp. 192-207

Résumé

Let $\Omega=[a,b]^\nu$ and let $T$ be a partially integral operator defined in $ L_2(\Omega^2)$ as follows: $$ (Tf)(x,y)=\int_\Omega q(x,s,y)f(s,y)\,d\mu(s). $$ In the article, we study the solvability of the partially integral Fredholm equations $f-\varkappa Tf=g$, where $g\in L_2(\Omega^2)$ is a given function and $\varkappa\in\mathbb C$. The notion of determinant (which is a measurable function on $\Omega$) is introduced for the operator $E-\varkappa T$, with $E$ is the identity operator in $L_2(\Omega^2)$. Some theorems on the spectrum of a bounded operator $T$ are proven.