Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Local and global factorizations of matrix-valued functions


Authors: K. F. Clancey and I. Gohberg
Journal: Trans. Amer. Math. Soc. 232 (1977), 155-167
MSC: Primary 47G05; Secondary 45E05
DOI: https://doi.org/10.1090/S0002-9947-1977-0454742-4
MathSciNet review: 0454742
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let C be a simple closed Liapounov contour in the complex plane and A an invertible $ n \times n$ matrix-valued function on C with bounded measurable entries. There is a well-known concept of factorization of the matrix function A relative to the Lebesgue space $ {L_p}(C)$. The notion of local factorization of A relative to $ {L_p}$ at a point $ {t_0}$ in C is introduced. It is shown that A admits a factorization relative to $ {L_p}(C)$ if and only if A admits a local factorization relative to $ {L_p}$ at each point $ {t_0}$ in C. Several problems connected with local factorizations relative to $ {L_p}$ are raised.


References [Enhancements On Off] (What's this?)

  • [1] 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, 87-103; English transl.; Amer. Math. Soc. Transl. (2) 102 (1973), 1-14. MR 41 #4246a; 48 #6. MR 0259609 (41:4246a)
  • [2] -, General theorems on the factorization of matrix-valued functions. II. Some tests and their consequences, Mat. Issled. 3 (1968), no. 3, 3-18; English transl., Amer. Math. Soc. Transl. (2) 102 (1973), 15-26. MR 41 #4246b; 48 #6. MR 0259610 (41:4246b)
  • [3] R. G. Douglas, Banach algebra techniques in operator theory, Academic Press, New York, 1972. MR 0361893 (50:14335)
  • [4] -, Local Toeplitz operators (preprint).
  • [5] R. G. Douglas and D. E. Sarason, Fredholm Toeplitz operators, Proc. Amer. Math. Soc. 26 (1970), 117-120. MR 41 #4275. MR 0259639 (41:4275)
  • [6] P. L. Duren, The theory of $ {H^p}$ spaces, Academic Press, New York, 1970. MR 42 #3552. MR 0268655 (42:3552)
  • [7] I. C. Gohberg, The factorization problem in normed rings, functions of isometric and symmetric operators and singular integral equations, Uspehi Mat. Nauk 19 (1964), no. 1 (115), 71-124 = Russian Math. Surveys 19 (1964), no. 1, 63-114. MR 29 #487. MR 0163184 (29:487)
  • [8] I. C. Gohberg and I. A. Feldman, Projections methods for solving Weiner-Hopf equations, English transl; Math. monographs, vol. 41, Amer. Math. Soc., Providence, R. I., 1974. MR 50 #8149. MR 0355675 (50:8149)
  • [9] 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 transl., Amer. Math. Soc. Transl. (2) 14 (1960), 217-287. MR 21 #1506; MR 22 #3954. MR 0113114 (22:3954)
  • [10] I. C. Gohberg and N. Ya Krupnik, Introudction to the theory of one-dimensional singular integral operators, ``Stiinca", Kishinev, 1973. (Russian) MR 0405177 (53:8971)
  • [11] G. M. Goluzin, Geometric theory of functions of a complex variable, English transl; Transl. Math. Monographs, vol. 26, Amer. Math. Soc., Providence, R. I., 1969. MR 40 #308. MR 0247039 (40:308)
  • [12] N. I. Mushelišvili, Singular integral equations. Boundary problems of function theory and their application to mathematical physics, 2nd ed., Fizmatgiz, Moscow, 1962; English transl. of 1st ed., Noordhoff, Groningen, 1953; reprinted, 1972. MR 15, 434; 50 #7968. MR 0355494 (50:7968)
  • [13] M. A. Shubin, On the local principle in the problem of factorization, Mat. Issled 6 (1971), no. 1, 174-180. MR 0285712 (44:2930)
  • [14] I. B. Simonenko, A new general method of investigating linear operator equations of the type of singular integral equations. I, Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965), 567-586. (Russian) MR 31 #3876. MR 0179630 (31:3876)
  • [15] -, Some general questions in the theory of Riemann boundary problems, Izv. Akad. Nauk SSSR 32 (1968) = Math USSR Izv. 2 (1968), 1091-1099.
  • [16] H. Röhrl, On holomorphic families of fiber bundles over the Riemann Sphere, Mem. Coll. Sci. Univ. Kyoto Ser. A Math. 33 (1960/61), 435-477. MR 24 #A1728. MR 0131881 (24:A1728)
  • [17] H. Widom, Singular integral equations in $ {L_p}$, Trans. Amer. Math. Soc. 97 (1960), 131-160. MR 22 #9830. MR 0119064 (22:9830)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 47G05, 45E05

Retrieve articles in all journals with MSC: 47G05, 45E05


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1977-0454742-4
Keywords: Operator factorizations, systems of singular integral equations
Article copyright: © Copyright 1977 American Mathematical Society

American Mathematical Society