On systems of linear functional equations of the second kind in $L_2$

We consider a general system of functional equations of the second kind in $L_2$ with a continuous linear operator $T$ satisfying the condition that zero lies in the limit spectrum of the adjoint operator $T^*$. We show that this condition holds for the operators of a wide class containing, in particular, all integral operators. The system under study is reduced by means of a unitary transformation to an equivalent system of linear integral equations of the second kind in$L_2$ with Carleman matrix kernel of a special kind. By a linear continuous invertible change, this system is reduced to an equivalent integral equation of the second kind in $L_2$ with quasidegenerate Carleman kernel. It is possible to apply various approximate methods of solution for such an equation.