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Proceedings of the American Mathematical Society

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

Author: Guillermo Miranda
Journal: Proc. Amer. Math. Soc. 26 (1970), 423-427
MSC: Primary 45.15
MathSciNet review: 0264349
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Abstract: 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 $ \vert\lambda \vert$ 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 $ \vert\lambda \vert < \omega $ is proved. In contrast to the Fredholm radius, an easy computation for $ \omega $ is available.

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Keywords: Singular systems, integral equations, Fredholm alternative, Fredholm radius, finite double-norm transformations, Neumann series, space of bounded functions
Article copyright: © Copyright 1970 American Mathematical Society