Stability of the eigenvalues of certain types of singular integral equations

In the paper one finds necessary and sufficient conditions for the existence of nontrivial solutions of the equation (A+C)f=f for all compact (or small in norm) perturbations C; for the unperturbed operators A one considers the operators of dual and conjugate to dual integral equations with kernels depending on the difference of arguments, Wiener-Hopf operators, and singular integral operators. The answer is given in terms of the symbol S (A) of the operator A.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Maternaticheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 138, pp. 127–136, 1984.