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 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 . The notion of local factorization of *A* relative to at a point in *C* is introduced. It is shown that *A* admits a factorization relative to if and only if *A* admits a local factorization relative to at each point in *C*. Several problems connected with local factorizations relative to are raised.

**[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. vyp. 2 (8), 87–103 (Russian). MR**0259609****[2]**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. vyp. 3 (9), 3–18 (Russian). MR**0259610****[3]**Ronald G. Douglas,*Banach algebra techniques in operator theory*, Academic Press, New York-London, 1972. Pure and Applied Mathematics, Vol. 49. MR**0361893****[4]**-,*Local Toeplitz operators*(preprint).**[5]**R. G. Douglas and Donald Sarason,*Fredholm Toeplitz operators*, Proc. Amer. Math. Soc.**26**(1970), 117–120. MR**0259639**, https://doi.org/10.1090/S0002-9939-1970-0259639-X**[6]**Peter L. Duren,*Theory of 𝐻^{𝑝} spaces*, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR**0268655****[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). MR**0163184****[8]**I. C. Gohberg and I. A. Fel′dman,*Convolution equations and projection methods for their solution*, American Mathematical Society, Providence, R.I., 1974. Translated from the Russian by F. M. Goldware; Translations of Mathematical Monographs, Vol. 41. MR**0355675****[9]**I. C. Gohberg and M. G. Kreĭn,*Systems of integral equations on a half line with kernels depending on the difference of arguments*, Amer. Math. Soc. Transl. (2)**14**(1960), 217–287. MR**0113114****[10]**I. Ts. Gokhberg and N. Ya. Krupnik,*\cyr Vvedenie v teoriyu odnomernykh singulyarnykh integral′nykh operatorov.*, Izdat. “Štiinca”, Kishinev, 1973 (Russian). MR**0405177****[11]**G. M. Goluzin,*Geometric theory of functions of a complex variable*, Translations of Mathematical Monographs, Vol. 26, American Mathematical Society, Providence, R.I., 1969. MR**0247039****[12]**N. I. Muskhelishvili,*Singular integral equations*, Wolters-Noordhoff Publishing, Groningen, 1972. Boundary problems of functions theory and their applications to mathematical physics; Revised translation from the Russian, edited by J. R. M. Radok; Reprinted. MR**0355494****[13]**M. A. Šubin,*The local principle in the factorization problem*, Mat. Issled.**6**(1971), no. vyp. 1 (19), 174–180 (Russian). MR**0285712****[14]**I. B. Simonenko,*A new general method of investigating linear operator equations of singular integral equation type. I*, Izv. Akad. Nauk SSSR Ser. Mat.**29**(1965), 567–586 (Russian). MR**0179630****[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]**Helmut Röhrl,*On holomorphic families of fiber bundles over the Riemannian sphere.*, Mem. Coll. Sci. Univ. Kyoto Ser. A Math.**33**(1960/1961), 435–477. MR**0131881**, https://doi.org/10.1215/kjm/1250711997**[17]**Harold Widom,*Singular integral equations in 𝐿_{𝑝}*, Trans. Amer. Math. Soc.**97**(1960), 131–160. MR**0119064**, https://doi.org/10.1090/S0002-9947-1960-0119064-7

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