On a transformation of integral equations

Abstract

Let E = E(a, b) be some Banach space of measurable functions on (a, b), I be the identity operator, and let $$\hat K$$ be a Fredholm-type regular integral operator acting on E and $${\hat K_ \pm }$$ be its triangular parts. We consider the representation $$I - \hat K = \left( {I - {{\hat K}_ - }} \right)\left( {I - \hat U} \right)\left( {I - {{\hat K}_ + }} \right)$$ , for some known classes of integral operators. In particular,we show that under certain conditions the operator $$\hat U$$ is positive and its spectral radius satisfies the condition $$r\left( {\hat U} \right) < 1$$ . Also, we give some possible applications of the representation.