Algebraic type of solutions for singular integral equations of the form \((S+T)x=x_ 0\) in Banach spaces (Q5905458)

The author studies singular integral equations of the form (*) \((S+T)x=x_ 0\) on the Hölder space \(H^ \alpha(L)\), where \(L\) is a closed curve in the complex plane, \((Sx)(t):=a(t)x(t)+(b(t)/\pi i)\) p.v. \(\int_ L{x(\tau)\over \tau-t}d\tau\) with \(a,b\in H^ \mu(L)(\alpha<\mu/2)\), \(a^ 2(t)-b^ 2(t)\neq 0\) and \((Tx)(t):=(1/\pi i)\int_ LT(t,\tau)x(\tau)d\tau\) with suitable kernel \(T(t,\tau)\). Necessary and sufficient conditions for the existence of a solution to (*) and the general solution (if it exists) are given, using previous results of the author.