Abstract:
We consider the factorization $I-K=(I-U^+)(I-U^-)$, where $I$ is the identity
operator, $K$ is an integral operator acting on some Banach space of functions
summable with respect to a measure $\mu$ on $(a,b)\subset(-\infty,+\infty)$
continuous relative to the Lebesgue measure,
\begin{equation*}
(Kf)(x)=\int^b_ak(x,t)f(t)\mu(dt),\qquad x\in(a,b),
\end{equation*}
and $U^\pm$ are the desired Volterra operators. A necessary and sufficient
condition is found for the existence of this factorization for a rather wide
class of operators $K$ with positive kernels and for Hilbert–Schmidt
operators.
Keywords:functions summable with respect to a measure, integral operators, Volterra factorization.
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