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The spectrum and eigenspaces of a meromorphic operator-valued function

Published online by Cambridge University Press:  14 November 2011

Robert Magnus
Robert Magnus
Affiliation:
The University Science Institute, Dunhaga 3, 107-Reykjavik., Iceland
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Synopsis

It is shown how to associate eigenvectors with a meromorphic mapping defined on a Riemann surface with values in the algebra of bounded operators on a Banach space. This generalises the case of classical spectral theory of a single operator. The consequences of the definition of the eigenvectors are examined in detail. A theorem is obtained which asserts the completeness of the eigenvectors whenever the Riemann surface is compact. Two technical tools are discussed in detail: Cauchy-kernels and Runge's Approximation Theorem for vector-valued functions.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 127 , Issue 5 , 1997 , pp. 1027 - 1051
Copyright
Copyright © Royal Society of Edinburgh 1997

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References

1Arason, Jón and Magnus, Robert. An algebraic multiplicity theory for analytic operator-valued functions (Preprint RH-01-94, Science Institute, University of Iceland, 1994). To appear in Math. Scand.Google Scholar
2Behnke, H. and Sommer, F.. Theorie der Analytischen Funktionen einer Komplexen Veränderlichen (Berlin: Springer, 1962).CrossRefGoogle Scholar
3Behnke, H. and Stein, K.. Entwicklungen analytischer Funktionen auf Riemannschen Flächen. Math. Ann. 120 (1948), 430–61.CrossRefGoogle Scholar
4Bliss, G. A.. Algebraic Functions (New York: Dover Publications, 1966; originally published 1933).Google Scholar
5Brown, L., Gauthier, P. M. and Seidel, W.. Complex approximation for vector-valued functions with an application to boundary behaviour. Trans. Amer. Math. Soc. 191 (1974), 149–64.CrossRefGoogle Scholar
6Farkas, H. M. and Kra, I.. Riemann Surfaces (New York: Springer, 1980).CrossRefGoogle Scholar
7Friedman, A. and Shinbrot, M.. Non-linear eigenvalue problems. Ada Math. 121 (1968), 77–125,Google Scholar
8Gauthier, P. M.. Analytic approximation on closed subsets of open Riemann surfaces. Constructive Function Theory 77. Sofia 1980, pp. 317–25.Google Scholar
9Gauthier, P. M.. Meromorphic uniform approximation on closed subsets of open Riemann surfaces. In Approximation Theory and Functional Analysis, ed. Prolla, J. B. (Amsterdam: North-Holland, 1979).Google Scholar
10Gauthier, P. M. and Rubel, L. A.. Holomorphic functional on open Riemann surfaces. Canad. J. Math. 28/ 4 (1976), 885–8.CrossRefGoogle Scholar
11Gohberg, I. C. and Sigal, E. I.. An operator generalization of the logarithmicresidue theorem and the theorem of Rouché. Math. USSR Sb. 13 (1971), No. 4 (English translation).CrossRefGoogle Scholar
12Gunning, R. C.. Lectures on Riemann Surfaces—Jacobi Varieties, Math. Notes 12 (Princeton: Princeton University Press, 1972).Google Scholar
13Keldys, M. V.. On the characteristic values and characteristic functions of certain classes of non-selfadjoint equations. Dokl. Akad. Nauk SSSR 77 (1951), 11–14 (in Russian).Google Scholar
14Köditz, H. and Timman, S.. Randschlichte meromorphe Funktionen auf endlichen Riemannschen Flächen. Math. Ann. 217 (1975) 157–9.CrossRefGoogle Scholar
15Magnus, R. J.. A generalization of multiplicity and the problem of bifurcation. Proc. London Math. Soc. 32 (1976), 251–78.CrossRefGoogle Scholar
16Magnus, R. J.. On the multiplicity of an analytic operator-valued function. Math. Scand. 77 (1995), 108–119.CrossRefGoogle Scholar
17Markus, A. S. and Sigal, E. I.. The multiplicity of the characteristic number of an operator valued function. Mat. Issled. 5 vyp. 3 (17) (1970), 129–47.Google Scholar
18Reed, M. and Simon, B.. Methods of Modern Mathematical Physics IV: Analysis ofOperators (New York: Academic Press, 1978).Google Scholar
19Scheinberg, S.. Uniform approximation by meromorphic functions having prescribed poles. Math. Ann. 243 (1979), 83–93.CrossRefGoogle Scholar
20Springer, G.. Introduction to Riemann Surfaces (Reading, Mass.: Addison-Wesley, 1957).Google Scholar