1. Section 1. The construction of the factorization of a triangular matrix function and the result in Proposition 1.2 is from Gohberg and Krein [26. I.C. Gohberg and M.G. Krein: Systems of integral equations on a half line with kernels depending on the difference of arguments. Uspehi Mat. Nauk, 13 (1958), no. 2(80), 3–72; English transi., Amer. Math. Soc. transi. (2)(1960), 217–287.
2. I.C. Gohberg and M.G. Krein: Systems of integral equations on a half line with kernels depending on the difference of arguments. Uspehi Mat. Nauk, 13 (1958), no. 2(80), 3–72; English transi., Amer. Math. Soc. transi. (2)(1960), 217–287].
3. Section 2. The rule developed in this section to obtain the partial indices of triangular 2 × 2-matrix functions is due to G. N. Chebotarev [11. G.N. Čebotarev: Partial indices for the Riemann boundary problem for a triangular matrix of second order. Uspehi Mat. Nauk 11(1956), 199–202. (Russian).
4. G.N. Čebotarev: Partial indices for the Riemann boundary problem for a triangular matrix of second order. Uspehi Mat. Nauk 11(1956), 199–202. (Russian)].
5. Section 3. The results in this section are from Gohberg and Lerer [36. I.C. Gohberg and L.E. Lerer: Singular integral operators as a generalization of the resultant matrix. Applicable Analysis, 7(1978),191–205].