Abstract
This chapter deals with various applications and results on generalized factorization of matrix functions. It is shown that any non-singular continuous matrix function admits a generalized factorization relative to any Lp(Γ) (D (l<p<∞). This result is contrasted to the case of non-singular piecewise continuous matrix functions which, in general, admit factorization relative to Lp(Γ) only for certain values of p and have partial indices depending on p . An application of generalized factorization to a basis problem in L2 is presented. Earlier results on factorization of dissipative and self-adjoint matrix functions are generalized. In the final section factorization relative to Lp(Γ) is considered as a separate case.
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Section 1. The resuit in Theorem 1.1 on generalized factorization of continuous matrix functions is from the book of Gohberg and Feldman I.C. Gohberg and I.A. Feldman: Convolution equations and projection methods for their solution. Transi. Math. Monographs, vol.41, Amer. Math. Soc., Providence, R.I., 1974. I.C. Gohberg and I.A. Feldman: Convolution equations and projection methods for their solution. Transi. Math. Monographs, vol.41, Amer. Math. Soc., Providence, R.I., 1974].
Section 2. The generalized factorization of piecewise continuous matrix functions has been handled in complete detail by Gohberg and Krupnik I. Gohberg and N. Ya. Krupnik: Systems of singular integral equations in weight spaces L. Dokl. Akad. Nauk. SSSR, 186 (1969), 998–1001.
Section 2. The generalized factorization of piecewise continuous matrix functions has been handled in complete detail by Gohberg and Krupnik I. Gohberg and N. Ya. Krupnik: English Transi. Soviet Math. Dokl. 10(1969), 688–691.
I. Gohberg and N. Ya. Krupnik: Systems of singular integral equations in weight spaces L . Dokl. Akad. Nauk. SSSR, 186 (1969), 998–1001.
I. Gohberg and N. Ya. Krupnik: English Transi. Soviet Math. Dokl. 10(1969), 688–691]. Further references and the details of the scalar case are in the book of Gohberg and Krupnik
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
I.C. Gohberg and M.G. Krein: Systems of integral equations on a half line with kernels depending on the difference of arguments English transi., Amer. Math. Soc. transi. (2)(1960), 217–287, Chap. [X].
Section 4. The result in Proposition 4.1 dealing with canonical generalized factorization relative to L2(Γ0) has been recently generalized to factorizations relative to LP(Γ0) by I. M. Spitkovskii H.H. Rosenbrock: State space and multivariable theory, Nelson, London, 1970.
H.H. Rosenbrock: State space and multivariable theory, Nelson, London, 1970]. The constancy of sgn A(t) a.e. obtained in Proposition 4.3 was first observed by H. R. Pousson
H.R. Pousson: Systems of Toeplitz operators on H. Proc. Amer. Math. Soc. 19(1968), 603–608].
Section 5. The main result of this section Theorem 5.1 is due to Rabindranathan M. Rabindranathan: On the inversion of Toeplitz operators. J. Math. Mech. 19(1969), 195–206 (see, also H. R. Pousson [58. H.R. Pousson: Systems of Toeplitz operators on H . Proc. Amer. Math. Soc. 19(1968), 603–608]). Lemma 5.1 is also from Rabindranathan’s paper. The scalar versions of Theorem 5.1 and Lemma 5.1 are due to Devinatz
A. Devinatz: Toeplitz Operators on H spaces. Trans. Amer. Math. Soc. 112(1964), 304–317] and Widom
H. Widom: Inversion of Toeplitz Matrices III. Notices Amer. Math. Soc. 7(1960), 63]. The lifting theorem of Sz. Nagy and Foias
B. Sz-Nagy and C. Foias: Dilation des commutante d’opera-teurs. C R. Acad. Sci. Paris, série A, 266(1968), 493–495] was first formulated in a special case by Sarason
D.E. Sarason: Generalized interpolation on H. Trans. Amer. Math. Soc. 127(1967), 179–203]. Further results on factorization and Toeplitz operators for the case p = 2 can be found in Douglas
R.G. Douglas: Banach algebra techniques in operator theory. Academic Press, New York and London, 1972
R.G. Douglas: Banach algebra techniques in the theory of Toeplitz operators. Conference Board of the Math. Sci., Regional Conference Ser. in Math., no. 15, Amer. Math. Soc, Providence, R.I., 1973] and Sarason. D.E. Sarason: Lecture notes on Toeplitz operators, University of Kentucky, 1970].
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Clancey, K.F., Gohberg, I. (1981). Further Results Concerning Generalized Factorization. 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_9
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