Abstract
In the paper Wiener-Hopf operators on a semigroup of nonnegative elements of a linearly quasi-ordered torsion free Abelian group are considered. Wiener-Hopf factorization of an invertible element of the group algebra is constructed, notions of a topological index and a factor index are introduced. It turns out that the set of factor indices for invertible elements of the group algebra is a linearly ordered group. It is shown that Wiener-Hopf operator with an invertible symbol is an one-side invertible operator and its invertibility properties are defined by the sign of the factor index of its symbol. Groups on which there exist nontrivial Fredholm Wiener-Hopf operators are described. As an example, all linear quasi-orders on the group ℤn are found and corresponding Wiener-Hopf operators are considered.
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References
Adukov, V.M.: Wiener-Hopf equations on a subsemigroup of a discrete Abelian group. I, VINITI AN SSSR, N2431-78, 1978 [Russian].
Arens, R. and Singer, I.M.: Generalized analytic functions, Trans Amer. Math. Soc.81 (1956), N 2, 379–393.
Coburn, L.A. and Douglas, R.G.: C*-algebra on halfspace. I, Inst. Hautes Etudes Sci. Publ. Math. 1971, N 40, 59–67.
Coburn, L.A., Douglas, R.G., Schaeffer, D.C. and Singer, I.M.: C*-algebra on half-space. II. Index Theory, Inst. Hautes Etudes Sci. Publ. Math. 1971, N 40, 69–79.
Fuchs, L.: Partially ordered algebraic systems, Pergamon Press, Oxford-London-New-York-Paris, 1963.
Fuchs, L.: Abelian groups, Akademiai Kiado, Budapest, 1966.
Gohberg, I.C. and Feldman, I.A.: Convolution equations and projection methods for their solutions, Amer. Math. Soc. Transl. Math. Monographs41, Providence, R.I., 1974.
Goldenstein, L.S. and Gohberg, I.C.: On a multivariate equation on a half-space with a kernel depending on the difference of arguments, Dokl. Akad. Nauk SSSR131 (1960), N1, 9–13 [Russian].
Hirschman, I.I.: Finite section Wiener-Hopf equations on a compact group with ordered dual, Bull. Amer. Math. Soc.70 (1964), N4, 508–510.
Hirschman, I.I.: Szego functions on a locally compact Abelian group with ordered dual, Trans. Amer. Math. Soc.121 (1966), N1, 133–159.
van Kampen, E.R.: On almost periodic function of constant absolute value, J. London Math. Soc.,12 (1937), N1, 3–6.
Krein, M.G.: Integral equations on the half-line with kernels depending on the difference of arguments, Amer. Math. Soc. Transl.22 (1962), 163–288.
Morris, S.A.: Pontryagin duality and the structure of locally compact Abelian groups, London Math. Soc. Lecture Note29 (1977).
Simonenko, I.B.: On multivariate discrete convolutions, Matem. Issled (Kishinev)3 (1968), N1, 108–122 [Russian].
Teh, H.H.: Construction of orders in Abelian groups, Proc. Cambridge Phil. Soc.57 (1961), N3, 476–482.
Zajtceva, M.I.: On the set of orderings of Abelian groups, Uspehi Mat. Nauk8 (1953), N1, 135–137 [Russian].
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Adukov, V. Wiener-Hopf operators on a subsemigroup of a discrete torsion free abelian group. Integr equ oper theory 16, 305–332 (1993). https://doi.org/10.1007/BF01204225
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DOI: https://doi.org/10.1007/BF01204225