Regularization of singular systems of integral equations with kernels of finite double-norm on $L_{\infty }$

There are known examples of linear integral transformations $T$ of finite double-norm on ${L_\infty }$ such that neither the transformation nor any of its iterates is compact, so that Fredholm’s alternative does not hold unrestrictedly for the equation $(I - \lambda T)g = f$ ($\lambda$ a complex number, $g,f \in {L_\infty }$. It is also known that the alternative holds true for $|\lambda |$ less than the Fredholm radius of $T$. Using a kernel decomposition, a quantity $\omega$ is introduced and the equivalence of an integral transformation system with components of finite double-norm on ${L_\infty }$, to a similar system that satisfies the Fredholm alternative for $|\lambda | < \omega$ is proved. In contrast to the Fredholm radius, an easy computation for $\omega$ is available.