An integral equation with matrix difference kernel on an interval

An equation of convolution type on an interval is considered. A realization of an extension of the corresponding integral operator in the form of an operator of Wiener-Hopf type is obtained. A result on the structure of the eigenspaces of the original operator and a criterion for its invertibility are proved on this basis. A formula enabling one to find the resolvent of the original operator given a factorization of the symbol of the auxiliary Wiener-Hopf operator is obtained.