Geodesic
On the factorization of integral operators on spaces of summable functions
Izvestiya. Mathematics , Tome 73 (2009) no. 5, pp. 921-937.

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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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N. B. Engibaryan. On the factorization of integral operators on spaces of summable functions. Izvestiya. Mathematics , Tome 73 (2009) no. 5, pp. 921-937. http://geodesic.mathdoc.fr/item/IM2_2009_73_5_a2/

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