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### The continuous invertibility of functional operators in Banach spaces

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Abstract: 1. Introduction. This paper is the survey of results on the one- and two-sided (continuous) invertibility for some classes of functional operators in Hölder, Lebesgue and Orlicz spaces, which were obtained in the theory of singular integral operators with discrete groups of shifts. In particular, we consider the invertibility in these spaces for binomial and polynomial operators generated by shift operators, local theory of invertibility in algebras of functional operators with discrete groups of shifts, the solvability of systems of difference equations with incommensurable differences on the semi-axis and finite intervals, and approximation approach to the problem of invertibility of such operators.

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- Hydroacoustics Branch, Marine Hydrophysical Institute, Ukrainian Academy of Sciences, Soviet Army St. 3, 270100 Odessa, Ukraine

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Bibliografia

[1] A. B. Antonevich, On operators generated by linear extensions of diffeomorphisms, Dokl. Akad. Nauk SSSR 243 (1978), 825-828. English transl. in Soviet Math. Dokl. 19 (1978).

[2] A. B. Antonevich, On two methods for investigation the invertibility of operators from C*-algebras generated by dynamical systems, Mat. Sb. 124 (1984), no. 5, 3-23. English transl. in Math. USSR Sb. 52 (1985).

[3] A. B. Antonevich, Linear Functional Equations: The Operator Approach, University Press, Minsk, 1988 (in Russian).

[4] V. D. Aslanov and Yu. I. Karlovich, One-sided invertibility of functional operators in reflexive Orlicz spaces, Dokl. Akad. Nauk AzSSR 45 (1989), no. 11-12, 3-7 (in Russian).

[5] R. G. Babadzhanyan and V. S. Rabinovich, On a factorization of almost periodic operator-valued functions, in: Differential, Integral Equations and Complex Analysis, Elista, 1986, 13-22 (in Russian).

[6] D. W. Boyd, Indices for the Orlicz spaces, Pacific J. Math. 38 (1971), no. 2, 315-323.

[7] A. O. Gel'fond, The Calculus of Finite Differences, 3rd ed., Nauka, Moscow, 1967 (in Russian).

[8] I. Ts. Gokhberg and I. A. Fel'dman, Convolution Equations and Projection Methods for Their Solution, Nauka, Moscow, 1971. English transl., Amer. Math. Soc., Providence, R.I., 1974.

[9] F. P. Greenleaf, Invariant Means on Topological Groups and Their Applications, Van Nostrand Reinhold, New York, 1969.

[10] Yu. I. Karlovich, On the invertibility of functional operators with non-Carleman shift in Hölder spaces, Differentsial'nye Uravneniya 20 (1984), 2165-2169 (in Russian).

[11] Yu. I. Karlovich, The local-trajectory method of studying invertibility in C*-algebras of operators with discrete groups of shifts, Dokl. Akad. Nauk SSSR 299 (1988), 546-550. English transl. in Soviet Math. Dokl. 37 (1988), 407-412.

[12] Yu. I. Karlovich, On algebras of singular integral operators with discrete groups of shifts in $L_p$-spaces, Dokl. Akad. Nauk SSSR 304 (1989), 274-280. English transl. in Soviet Math. Dokl. 39 (1989), 48-53.

[13] Yu. I. Karlovich, Riemann and Haseman vector boundary value problems with oscillating coefficients, Dokl. Rasshir. Zased. Semin. IPM im. I. N. Vekua, Tbilisi 5 (1990), no. 1, 86-89 (in Russian).

[14] Yu. I. Karlovich, C*-algebras of nonlocal quaternionic convolution type operators, in: Clifford Algebras and their Applications in Mathematical Physics: Proc. of the Third Conference held at Deinze, Belgium, 1993, Kluwer Acad. Publ., Dordrecht, 1993, 109-118.

[15] Yu. I. Karlovich and V. G. Kravchenko, A Noether theory for a singular integral operator with a shift having periodic points, Dokl. Akad. Nauk SSSR 231 (1976), 277-280. English transl. in Soviet Math. Dokl. 17 (1976), 1547-1551.

[16] Yu. I. Karlovich and V. G. Kravchenko, On systems of functional and integro-functional equations with a non-Carleman shift, Dokl. Akad. Nauk SSSR 236 (1977), 1064-1067. English transl. in Soviet Math. Dokl. 18 (1977), 1319-1322.

[17] Yu. I. Karlovich and V. G. Kravchenko, On an algebra of singular integral operators with non-Carleman shift, Dokl. Akad. Nauk SSSR 239 (1978), 38-41. English transl. in Soviet Math. Dokl. 19 (1978), 267-271.

[18] Yu. I. Karlovich and V. G. Kravchenko, On some new results in the Noether theory of singular integral operators with non-Carleman shift, in: Sovrem. Probl. Teor. Funk., Baku, 1980, 145-150 (in Russian).

[19] Yu. I. Karlovich and V. G. Kravchenko, Systems of singular integral equations with a shift, Mat. Sb. 116 (1981), no. 1, 87-110. English transl. in Math. USSR Sb. 44 (1983), no. 1, 75-95.

[20] Yu. I. Karlovich and V. G. Kravchenko, An algebra of singular integral operators with piecewise-continuous coefficients and a piecewise-smooth shift on a composite contour, Izv. Akad. Nauk SSSR Ser. Mat. 47 (1983), 1030-1077. English transl. in Math. USSR Izv. 23 (1984), no. 2, 307-352.

[21] Yu. I. Karlovich, V. G. Kravchenko and G. S. Litvinchuk, The invertibility of functional operators on Banach spaces, in: Funktsional-Differ. Uravn., Perm, 1990, 18-58 (in Russian).

[22]Yu. I. Karlovich, V. G. Kravchenko and G. S. Litvinchuk, On Noethericity and Mikhlin symbols of operators of the type of singular integral operators with shift, Z. Anal. Anwend. 9 (1990), 15-32 (in Russian).

[23] Yu. I. Karlovich, Yu. D. Latushkin and R. Mardiev, On one-sided invertibility of functional operators and n(d)-normality of singular integral operators with a shift, Odessa, 1984, 29 p. Manuscript no. 8361-84, deposited at VINITI (in Russian).

[24] Yu. I. Karlovich, Yu. D. Latushkin and R. Mardiev, Criterion for n(d)-normality of singular integral operators with non-Carleman shift, in: Funktsional-Differ. Uravn., Perm, 1985, 45-50 (in Russian).

[25] Yu. I. Karlovich and G. S. Litvinchuk, Algebras of singular integral operators with discrete groups of shifts, in: Sovrem. Probl. Mat. Fiz., Tbilisi 2 (1987), 57-64 (in Russian).

[26] Yu. I. Karlovich and G. S. Litvinchuk, On some classes of semi-Noetherian operators, Izv. Vyssh. Uchebn. Zaved. Mat. (1990), no. 2, 3-16 (in Russian).

[27] Yu. I. Karlovich and R. Mardiev, On one-sided invertibility of functional operators with non-Carleman shift in Hölder spaces, Izv. Vyssh. Uchebn. Zaved. Mat. (1987), no. 3, 77-80 (in Russian).

[28] Yu. I. Karlovich and R. Mardiev, One-sided invertibility of functional operators and the n(d)-normality of singular integral operators with translation in Hölder spaces, Differentsial'nye Uravneniya 24 (1988), 488-499. English transl. in Differential Equations 24 (1988), no. 3, 350-359.

[29] Yu. I. Karlovich and R. Mardiev, On one-sided invertibility of functional operators and n(d)-normality of singular integral operators with a shift having periodic points in Hölder spaces, Samarkand, 1988, 56 p. Manuscript no. 822-Uz88, deposited at UzNIINTI (in Russian).

[30] Yu. I. Karlovich and R. Mardiev, On n(d)-normality of singular integral operators with a shift in Hölder spaces, Dokl. Akad. Nauk UzSSR (1990), no. 4, 10-12 (in Russian).

[31] Yu. I. Karlovich and I. M. Spitkovskiĭ, On the Noetherian property of certain singular integral operators with matrix coefficients of class SAP and systems of convolution equations on a finite interval connected with them, Dokl. Akad. Nauk SSSR 269 (1983), 531-535. English transl. in Soviet Math. Dokl. 27 (1983), 358-363.

[32] Yu. I. Karlovich and I. M. Spitkovskiĭ, Factorization problem for almost periodic matrix-functions and Fredholm theory of Toeplitz operators with semi-almost periodic matrix symbols, in: Lecture Notes in Math. 1043, Springer, 1984, 279-282.

[33] Yu. I. Karlovich and I. M. Spitkovskiĭ, Factorization of almost periodic matrix-valued functions and (semi) Fredholmness of certain classes of convolution type equations, Odessa, 1985, 138 p. Manuscript no. 4421-85, deposited at VINITI (in Russian).

[34] Yu. I. Karlovich and I. M. Spitkovskiĭ, On the theory of systems of convolution type equations with semi-almost-periodic symbols in spaces of Bessel potentials, Dokl. Akad. Nauk SSSR 286 (1986), 799-803. English transl. in Soviet Math. Dokl. 33 (1986), 180-184.

[35] Yu. I. Karlovich and I. M. Spitkovskiĭ, Factorization of almost periodic matrix-valued functions and the Noether theory for certain classes of equations of convolution type, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), 276-308. English transl. in Math. USSR Izv. 34 (1990), 281-316.

[36] M. A. Krasnosel'skiĭ and Ya. B. Rutitskiĭ, Convex Functions and Orlicz Spaces, Fizmatgiz, Moscow, 1958 (in Russian).

[37] V. G. Kravchenko, On a singular integral operator with a shift, Dokl. Akad. Nauk SSSR 215 (1974), 1301-1304. English transl. in Soviet Math. Dokl. 15 (1974), 690-694.

[38] V. G. Kravchenko, On a functional equation with a shift in the space of continuous functions, Mat. Zametki 22 (1977), no. 2, 303-311 (in Russian).

[39] V. G. Kurbatov, Linear Differential-Difference Equations, University Press, Voronezh, 1990 (in Russian).

[40] Yu. D. Latushkin and A. M. Stepin, Weighted composition operators, spectral theory of linear extensions and multiplicative ergodic theorem, Mat. Sb. 181 (1990), no. 6, 723-742 (in Russian).

[41] Yu. D. Latushkin and A. M. Stepin, Weighted composition operators and linear extensions of dynamical systems, Uspekhi Mat. Nauk 46 (1991), no. 2, 85-143. English transl. in Russian Math. Surveys 46 (1992), 95-165.

[42] A. V. Lebedev, The invertibility of elements in the C*-algebras qenerated by dynamical systems, Uspekhi Mat. Nauk 34 (1979), no. 4, 199-200. English transl. in Russian Math. Surveys 34 (1979).

[43] R. Mardiev, A criterion for the semi-Noetherian property of one class of singular integral operators with a non-Carleman shift, Dokl. Akad. Nauk UzSSR (1985), no. 2, 5-7 (in Russian).

[44] R. Mardiev, A criterion for n(d)-normality of singular integral operators with a shift having periodic points in Lebesgue spaces, Samarkand, 1988, 41 p. Manuscript no. 821-Uz88, deposited at UzNIINTI (in Russian).

[45] A. G. Myasnikov and L. I. Sazonov, On singular integral operators with non-Carleman shift, Dokl. Akad. Nauk SSSR 237 (1977), 1289-1292. English transl. in Soviet Math. Dokl. 18 (1977).

[46] A. G. Myasnikov and L. I. Sazonov, Singular integral operators with a non-Carleman shift, Izv. Vyssh. Uchebn. Zaved. Matematika (1980), no. 3 (214), 22-31. English transl. in Soviet Math. (Iz.VUZ) 24 (1980).

[47] A. G. Myasnikov and L. I. Sazonov, On singular operators with a non-Carleman shift and their symbols, Dokl. Akad. Nauk SSSR 254 (1980), 1076-1080. English transl. in Soviet Math. Dokl. 22 (1980).

[48] N. G. Samko, Singular integral operators with discontinuous coefficients on generalized Hölder spaces, Ph. D. dissertation, Rostov-on-Don, 1991 (in Russian).

[49] V. N. Semenyuta, On singular operator equations with shift on a circle, Dokl. Akad. Nauk SSSR 237 (1977), 1301-1302. English transl. in Soviet Math. Dokl. 18 (1977), 1572-1574.

[50] I. M. Spitkovskiĭ, Factorization of several classes of semi-almost periodic matrix functions and applications to systems of convolution equations, Izv. Vyssh. Uchebn. Zaved. Matematika (1983), no. 4, 88-94. English transl. in Soviet Math. (Iz.VUZ) 27 (1983), 383-388.

[51] I. M. Spitkovskiĭ, On the factorization of almost periodic matrix functions, Mat. Zametki 45 (1989), no. 6, 74-82. English transl. in Math. Notes 45 (1989), no. 5-6, 482-488.

[52] I. M. Spitkovskiĭ and P. M. Tishin, Factorization of new classes of almost periodic matrix functions, Dokl. Rasshir. Zased. Semin. IPM im. I. N. Vekua, Tbilisi, 3 (1989), no. 1, 170-173 (in Russian).

[2] A. B. Antonevich, On two methods for investigation the invertibility of operators from C*-algebras generated by dynamical systems, Mat. Sb. 124 (1984), no. 5, 3-23. English transl. in Math. USSR Sb. 52 (1985).

[3] A. B. Antonevich, Linear Functional Equations: The Operator Approach, University Press, Minsk, 1988 (in Russian).

[4] V. D. Aslanov and Yu. I. Karlovich, One-sided invertibility of functional operators in reflexive Orlicz spaces, Dokl. Akad. Nauk AzSSR 45 (1989), no. 11-12, 3-7 (in Russian).

[5] R. G. Babadzhanyan and V. S. Rabinovich, On a factorization of almost periodic operator-valued functions, in: Differential, Integral Equations and Complex Analysis, Elista, 1986, 13-22 (in Russian).

[6] D. W. Boyd, Indices for the Orlicz spaces, Pacific J. Math. 38 (1971), no. 2, 315-323.

[7] A. O. Gel'fond, The Calculus of Finite Differences, 3rd ed., Nauka, Moscow, 1967 (in Russian).

[8] I. Ts. Gokhberg and I. A. Fel'dman, Convolution Equations and Projection Methods for Their Solution, Nauka, Moscow, 1971. English transl., Amer. Math. Soc., Providence, R.I., 1974.

[9] F. P. Greenleaf, Invariant Means on Topological Groups and Their Applications, Van Nostrand Reinhold, New York, 1969.

[10] Yu. I. Karlovich, On the invertibility of functional operators with non-Carleman shift in Hölder spaces, Differentsial'nye Uravneniya 20 (1984), 2165-2169 (in Russian).

[11] Yu. I. Karlovich, The local-trajectory method of studying invertibility in C*-algebras of operators with discrete groups of shifts, Dokl. Akad. Nauk SSSR 299 (1988), 546-550. English transl. in Soviet Math. Dokl. 37 (1988), 407-412.

[12] Yu. I. Karlovich, On algebras of singular integral operators with discrete groups of shifts in $L_p$-spaces, Dokl. Akad. Nauk SSSR 304 (1989), 274-280. English transl. in Soviet Math. Dokl. 39 (1989), 48-53.

[13] Yu. I. Karlovich, Riemann and Haseman vector boundary value problems with oscillating coefficients, Dokl. Rasshir. Zased. Semin. IPM im. I. N. Vekua, Tbilisi 5 (1990), no. 1, 86-89 (in Russian).

[14] Yu. I. Karlovich, C*-algebras of nonlocal quaternionic convolution type operators, in: Clifford Algebras and their Applications in Mathematical Physics: Proc. of the Third Conference held at Deinze, Belgium, 1993, Kluwer Acad. Publ., Dordrecht, 1993, 109-118.

[15] Yu. I. Karlovich and V. G. Kravchenko, A Noether theory for a singular integral operator with a shift having periodic points, Dokl. Akad. Nauk SSSR 231 (1976), 277-280. English transl. in Soviet Math. Dokl. 17 (1976), 1547-1551.

[16] Yu. I. Karlovich and V. G. Kravchenko, On systems of functional and integro-functional equations with a non-Carleman shift, Dokl. Akad. Nauk SSSR 236 (1977), 1064-1067. English transl. in Soviet Math. Dokl. 18 (1977), 1319-1322.

[17] Yu. I. Karlovich and V. G. Kravchenko, On an algebra of singular integral operators with non-Carleman shift, Dokl. Akad. Nauk SSSR 239 (1978), 38-41. English transl. in Soviet Math. Dokl. 19 (1978), 267-271.

[18] Yu. I. Karlovich and V. G. Kravchenko, On some new results in the Noether theory of singular integral operators with non-Carleman shift, in: Sovrem. Probl. Teor. Funk., Baku, 1980, 145-150 (in Russian).

[19] Yu. I. Karlovich and V. G. Kravchenko, Systems of singular integral equations with a shift, Mat. Sb. 116 (1981), no. 1, 87-110. English transl. in Math. USSR Sb. 44 (1983), no. 1, 75-95.

[20] Yu. I. Karlovich and V. G. Kravchenko, An algebra of singular integral operators with piecewise-continuous coefficients and a piecewise-smooth shift on a composite contour, Izv. Akad. Nauk SSSR Ser. Mat. 47 (1983), 1030-1077. English transl. in Math. USSR Izv. 23 (1984), no. 2, 307-352.

[21] Yu. I. Karlovich, V. G. Kravchenko and G. S. Litvinchuk, The invertibility of functional operators on Banach spaces, in: Funktsional-Differ. Uravn., Perm, 1990, 18-58 (in Russian).

[22]Yu. I. Karlovich, V. G. Kravchenko and G. S. Litvinchuk, On Noethericity and Mikhlin symbols of operators of the type of singular integral operators with shift, Z. Anal. Anwend. 9 (1990), 15-32 (in Russian).

[23] Yu. I. Karlovich, Yu. D. Latushkin and R. Mardiev, On one-sided invertibility of functional operators and n(d)-normality of singular integral operators with a shift, Odessa, 1984, 29 p. Manuscript no. 8361-84, deposited at VINITI (in Russian).

[24] Yu. I. Karlovich, Yu. D. Latushkin and R. Mardiev, Criterion for n(d)-normality of singular integral operators with non-Carleman shift, in: Funktsional-Differ. Uravn., Perm, 1985, 45-50 (in Russian).

[25] Yu. I. Karlovich and G. S. Litvinchuk, Algebras of singular integral operators with discrete groups of shifts, in: Sovrem. Probl. Mat. Fiz., Tbilisi 2 (1987), 57-64 (in Russian).

[26] Yu. I. Karlovich and G. S. Litvinchuk, On some classes of semi-Noetherian operators, Izv. Vyssh. Uchebn. Zaved. Mat. (1990), no. 2, 3-16 (in Russian).

[27] Yu. I. Karlovich and R. Mardiev, On one-sided invertibility of functional operators with non-Carleman shift in Hölder spaces, Izv. Vyssh. Uchebn. Zaved. Mat. (1987), no. 3, 77-80 (in Russian).

[28] Yu. I. Karlovich and R. Mardiev, One-sided invertibility of functional operators and the n(d)-normality of singular integral operators with translation in Hölder spaces, Differentsial'nye Uravneniya 24 (1988), 488-499. English transl. in Differential Equations 24 (1988), no. 3, 350-359.

[29] Yu. I. Karlovich and R. Mardiev, On one-sided invertibility of functional operators and n(d)-normality of singular integral operators with a shift having periodic points in Hölder spaces, Samarkand, 1988, 56 p. Manuscript no. 822-Uz88, deposited at UzNIINTI (in Russian).

[30] Yu. I. Karlovich and R. Mardiev, On n(d)-normality of singular integral operators with a shift in Hölder spaces, Dokl. Akad. Nauk UzSSR (1990), no. 4, 10-12 (in Russian).

[31] Yu. I. Karlovich and I. M. Spitkovskiĭ, On the Noetherian property of certain singular integral operators with matrix coefficients of class SAP and systems of convolution equations on a finite interval connected with them, Dokl. Akad. Nauk SSSR 269 (1983), 531-535. English transl. in Soviet Math. Dokl. 27 (1983), 358-363.

[32] Yu. I. Karlovich and I. M. Spitkovskiĭ, Factorization problem for almost periodic matrix-functions and Fredholm theory of Toeplitz operators with semi-almost periodic matrix symbols, in: Lecture Notes in Math. 1043, Springer, 1984, 279-282.

[33] Yu. I. Karlovich and I. M. Spitkovskiĭ, Factorization of almost periodic matrix-valued functions and (semi) Fredholmness of certain classes of convolution type equations, Odessa, 1985, 138 p. Manuscript no. 4421-85, deposited at VINITI (in Russian).

[34] Yu. I. Karlovich and I. M. Spitkovskiĭ, On the theory of systems of convolution type equations with semi-almost-periodic symbols in spaces of Bessel potentials, Dokl. Akad. Nauk SSSR 286 (1986), 799-803. English transl. in Soviet Math. Dokl. 33 (1986), 180-184.

[35] Yu. I. Karlovich and I. M. Spitkovskiĭ, Factorization of almost periodic matrix-valued functions and the Noether theory for certain classes of equations of convolution type, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), 276-308. English transl. in Math. USSR Izv. 34 (1990), 281-316.

[36] M. A. Krasnosel'skiĭ and Ya. B. Rutitskiĭ, Convex Functions and Orlicz Spaces, Fizmatgiz, Moscow, 1958 (in Russian).

[37] V. G. Kravchenko, On a singular integral operator with a shift, Dokl. Akad. Nauk SSSR 215 (1974), 1301-1304. English transl. in Soviet Math. Dokl. 15 (1974), 690-694.

[38] V. G. Kravchenko, On a functional equation with a shift in the space of continuous functions, Mat. Zametki 22 (1977), no. 2, 303-311 (in Russian).

[39] V. G. Kurbatov, Linear Differential-Difference Equations, University Press, Voronezh, 1990 (in Russian).

[40] Yu. D. Latushkin and A. M. Stepin, Weighted composition operators, spectral theory of linear extensions and multiplicative ergodic theorem, Mat. Sb. 181 (1990), no. 6, 723-742 (in Russian).

[41] Yu. D. Latushkin and A. M. Stepin, Weighted composition operators and linear extensions of dynamical systems, Uspekhi Mat. Nauk 46 (1991), no. 2, 85-143. English transl. in Russian Math. Surveys 46 (1992), 95-165.

[42] A. V. Lebedev, The invertibility of elements in the C*-algebras qenerated by dynamical systems, Uspekhi Mat. Nauk 34 (1979), no. 4, 199-200. English transl. in Russian Math. Surveys 34 (1979).

[43] R. Mardiev, A criterion for the semi-Noetherian property of one class of singular integral operators with a non-Carleman shift, Dokl. Akad. Nauk UzSSR (1985), no. 2, 5-7 (in Russian).

[44] R. Mardiev, A criterion for n(d)-normality of singular integral operators with a shift having periodic points in Lebesgue spaces, Samarkand, 1988, 41 p. Manuscript no. 821-Uz88, deposited at UzNIINTI (in Russian).

[45] A. G. Myasnikov and L. I. Sazonov, On singular integral operators with non-Carleman shift, Dokl. Akad. Nauk SSSR 237 (1977), 1289-1292. English transl. in Soviet Math. Dokl. 18 (1977).

[46] A. G. Myasnikov and L. I. Sazonov, Singular integral operators with a non-Carleman shift, Izv. Vyssh. Uchebn. Zaved. Matematika (1980), no. 3 (214), 22-31. English transl. in Soviet Math. (Iz.VUZ) 24 (1980).

[47] A. G. Myasnikov and L. I. Sazonov, On singular operators with a non-Carleman shift and their symbols, Dokl. Akad. Nauk SSSR 254 (1980), 1076-1080. English transl. in Soviet Math. Dokl. 22 (1980).

[48] N. G. Samko, Singular integral operators with discontinuous coefficients on generalized Hölder spaces, Ph. D. dissertation, Rostov-on-Don, 1991 (in Russian).

[49] V. N. Semenyuta, On singular operator equations with shift on a circle, Dokl. Akad. Nauk SSSR 237 (1977), 1301-1302. English transl. in Soviet Math. Dokl. 18 (1977), 1572-1574.

[50] I. M. Spitkovskiĭ, Factorization of several classes of semi-almost periodic matrix functions and applications to systems of convolution equations, Izv. Vyssh. Uchebn. Zaved. Matematika (1983), no. 4, 88-94. English transl. in Soviet Math. (Iz.VUZ) 27 (1983), 383-388.

[51] I. M. Spitkovskiĭ, On the factorization of almost periodic matrix functions, Mat. Zametki 45 (1989), no. 6, 74-82. English transl. in Math. Notes 45 (1989), no. 5-6, 482-488.

[52] I. M. Spitkovskiĭ and P. M. Tishin, Factorization of new classes of almost periodic matrix functions, Dokl. Rasshir. Zased. Semin. IPM im. I. N. Vekua, Tbilisi, 3 (1989), no. 1, 170-173 (in Russian).

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