Abstract
In this chapter a study is made of matrix algebras of functions on a contour such that non-singular elements in these matrix algebras admit factorization relative to the contour with factors also belonging to the algebra. Such factorizations lead in a natural way to the concept of decomposing algebras of matrix functions. A detailed list of examples of decomposing algebras of functions is provided in Section 5. Connections are drawn between factorization and barrier problems in complex function theory.
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Notes
Section 1. The material in this section is well known. The exposition here is close to the description in Gohberg and Krupnik [28. I. Gohberg and N. Ya. Krupnik: Introduction to the theory of one-dimensional singular integral operators. Kishniev: Stiince, 1973(Russian) . German transi: Birkhauser,Verlag, Basel-Boston-Stuggart, 1979.]. For further references,
see Budjann and Gohberg [8. M.S. Budjanu and I.C. Gohberg: General theorems on the factorization of matrix valued functions, I. The fundamental theorem. Mat. Issled. 3(1968), no. 2(8), 87–103;
M.S. Budjanu and I.C. Gohberg: English transi., Amer. Math. Soc. Transi. (2) 102(1973), 1–14,
M.S. Budjanu and I.C. Gohberg: General theorems on the factorization of matrix-valued functions, II. Some tests and their consequences. Mat. Issled. 3(1968), no. 3(9), 3–18. English transi. Amer. Math. Soc. Transi. (2) 102 (1973), 5–26] and Atkinson
F.V. Atkinson: Some aspects of Baxter’s functional equation. J. Math. Anal. Appl. 7(1963), 1–30].
Section 2. The problem of factorization of scalar functions is discussed in the book of Gohberg and Krupnik [28. I. Gohberg and N. Ya. Krupnik: Introduction to the theory of one-dimensional singular integral operators. Kishniev: Stiince, 1973(Russian) . German transi: Birkhauser,Verlag, Basel-Boston-Stuggart, 1979., Chapter III]. The exposition here is a modification of the description in this book.
Section 3. The main theorem on factorization in decomposing R-algebras is due to Gohberg. See e.g.Budjanu and Gohberg [8. M.S. Budjanu and I.C. Gohberg: General theorems on the factorization of matrix valued functions, I. The fundamental theorem. Mat. Issled. 3(1968), no. 2(8), 87–103; English transi., Amer. Math. Soc. Transi. (2) 102(1973), 1–14,
M.S. Budjanu and I.C. Gohberg: General theorems on the factorization of matrix-valued functions, II. Some tests and their consequences. Mat. Issled. 3(1968), no. 3(9), 3–18.
M.S. Budjanu and I.C. Gohberg: English transi. Amer. Math. Soc. Transi. (2) 102 (1973), 5–26].
Section 4, 5. The results presented here are due to Budjanu and Gohberg [8. M.S. Budjanu and I.C. Gohberg: General theorems on the factorization of matrix valued functions, I. The fundamental theorem. Mat. Issled. 3(1968), no. 2(8), 87–103; English transi., Amer. Math. Soc. Transi. (2) 102(1973), 1–14,
M.S. Budjanu and I.C. Gohberg: General theorems on the factorization of matrix-valued functions, II. Some tests and their consequences. Mat. Issled. 3(1968), no. 3(9), 3–18. English transi. Amer. Math. Soc. Transi. (2) 102 (1973), 5–26], We mention that condition 2(b) in Theorem 4.1 can be replaced by the following:
Further, in general, it is not known whether this spectral radius condition or 2(b) automatically holds in case C is a decomposing algebra.
Section 6. The results in Theorems 6.1 and 6.3 are 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] and Theorem 6.2 is from Muskhelishvili
N.I. Muskhelishvili: Singular integral equations. Boundary problems of function theory and their applications to mathematical physics. 2nd Ed., Fizmatgiz, Moscow, 1962; English transi, of 1st. ed., Noordhoff, Groningen, 1953 and Plemelj
J. Plemelj: Riemannsche Funktionenscharen mit gegebiner Monodromiegruppe. Monatsheft für Math, und Phys. XIX (1908), 221–245].
Section 7. This material is in the spirit of Muskelishvili [55. N.I. Muskhelishvili: Singular integral equations. Boundary problems of function theory and their applications to mathematical physics. 2nd Ed., Fizmatgiz, Moscow, 1962; English transi, of 1st. ed., Noordhoff, Groningen, 1953] and Vekua
N.P. Vekua: Systems of singular integral equations and some boundary problems. GITTL, Moscow, 1950. English transi., Noordhoff, Groningen, 1967] .
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Clancey, K.F., Gohberg, I. (1981). Decomposing Algebras of Matrix Functions. In: Factorization of Matrix Functions and Singular Integral Operators. Operator Theory: Advances and Applications, vol 3. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5492-4_3
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