Representation formula for solution of a functional equation with Volterra operator

The following functional equation is under consideration,equation(0.1)Lx=fLx=f with a linear continuous operator LL, defined on the Banach space X0(Ω0,Σ0,μ0;Y0)X0(Ω0,Σ0,μ0;Y0) of functions x0:Ω0→Y0 and having values in the Banach space X2(Ω2,Σ2,μ2;Y2)X2(Ω2,Σ2,μ2;Y2) of functions x2:Ω2→Y2. The peculiarity of X0X0 is that the convergence of a sequence xn0∈X0, n=1,2,…, to the function x0∈X0x0∈X0 in the norm of X0X0 implies the convergence xn0(s)→x0(s), s∈Ω0s∈Ω0, μ0μ0-almost everywhere. The assumption on the space X2X2 is that it is an ideal space. The suggested representation of solution to (0.1) is based on a notion of the Volterra   property together with a special presentation of the equation using an isomorphism between X0X0 and the direct product X1(Ω1,Σ1,μ1;Y1)×Y0X1(Ω1,Σ1,μ1;Y1)×Y0 (here X1(Ω1,Σ1,μ1;Y1)X1(Ω1,Σ1,μ1;Y1) is the Banach space of measurable functions x1:Ω1→Y1). The representation X0=X1×Y0X0=X1×Y0 leads to a decomposition of L:X0→X2 for the pair of operators Q:X1→X2 and A:Y0→X2. A series of basic properties of (0.1) is implied by the properties of operator Q.