# Inverse of Identity plus Volterra operator

consider the following operator or $L_2(0,1)$, $(Pw)(x)=w(x)+\int_0^x K(x,y)w(y)dy+\int_x^1 K(y,x)w(y)dy$, where the integral kernel is a polynomial.

I am trying to construct the inverse of this operator, I can construct inverses of the operators where the integral kernel is just a function of $y$, but I don't know how to proceed for the operator $P$. Note that the operator $P$ is self-adjoint.

Any help would be most helpful, thanks.

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Hi Mhenni, The kernel is not self-adjoint but the operator is. In fact $K(x,y) \neq K(y,x)$ for my case. A change of order of integration will prove the self-adjointedness of the operator $P$ without needing the property $K(x,y)=K(y,x)$. –  Amit Apr 3 '13 at 13:32
Additionally, the kernel is real. –  Amit Apr 3 '13 at 13:33
Sorry, absent minded. –  Mhenni Benghorbal Apr 3 '13 at 13:47