most recent 30 from http://mathoverflow.net 2013-05-21T19:10:49Z http://mathoverflow.net/feeds/question/88611 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/88611/decomposition-of-an-integral-operator-into-a-composition Victor Liu 2012-02-16T09:04:51Z 2012-02-16T09:04:51Z

<p>I've been musing about the following question for a while now. Given an integral operator $G$ defined by $$ (Gf)(x) = \int_0^1 G(x,u) f(u)\,du $$ Is it possible to decompose this into two separate "one-sided" integral operators $L$ and $R$ such that $$ (Gf)(x) = \int_x^1 L(x,u) \int_0^x R(u,v) f(v)\,dv\,du $$ If so, under what assumptions on $G$? And are there expressions for the left and right operators $L$ and $R$ in terms of $G$?</p>