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Fredholm Toeplitz Operators and Slow Oscillation

Published online by Cambridge University Press:  20 November 2018

S. C. Power
S. C. Power*
Affiliation:
University of Lancaster, Lancaster, England
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The purpose of this paper is to show how Fred hoi m criteria for Toeplitz operators, whose symbols lie in an algebra,A, may often be generalized to cover a larger symbol algebra generated by A and SO, the slowly oscillating functions. Mere A and SO are algebras of continuous functions on the real line, so that we are concerned principally with the effect of a single discontinuity in the symbol function.

We shall treat the cases when A is the almost periodic functions, the semi-almost periodic functions and the multiplicatively periodic functions. Sufficient criteria for Fredholmness are obtained in Section 5. The more difficult task of establishing necessary and sufficient criteria is only achieved here for the slowly oscillating almost periodic functions and this is done in Section 6.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 5 , 01 October 1980 , pp. 1058 - 1071
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. Abrahamse, M. B., The spectrum of a Toeplitz operator with a multiplicatively periodic symbol, preprint.CrossRefGoogle Scholar
2. Coburn, L. A. and Douglas, R. G., Translation operators of the half-line, Proc. Nat. Acad. Sci. U.S.A. 62 (1969), 1010–1013.Google Scholar
3. Devinatz, A., Toeplitz operators on H2 spaces, Trans. Amer. Math. Soc. 112 (1904), 304–317.Google Scholar
4. Douglas, R. G., Banach algebra techniques in operator theory (Academic Press, New-York and London, 1972).Google Scholar
5. Douglas, R. G., Banach algebra techniques in the theory of Toeplitz operators, C.B.M.S. Regional Conference no. 15 (Amer. Math. Soc, Providence, Rhode Island, 1973).CrossRefGoogle Scholar
6. Douglas, R. G., On the C*algebra of a one-parameter semi-group of isometrices, ACTA Math. 128 (1972), 143–151.Google Scholar
7. Douglas, R. G., Local Toeplitz operators, Proc. London Math. Soc. 36 (1978), 243–272.Google Scholar
8. Douglas, R. G. and Sarason, D. E., Fredholm Toeplitz operators, Proc. Amer. Math. Soc. 26 (1970), 117–120.Google Scholar
9. Gohberg, I. C. and Feldman, I. A., Weiner-Hopf integral difference equations, ACTA Sci. Math. 30 (1969), 199–224.Google Scholar
10. Gohberg, I. C. and Kruprik, N. Ja., The algebra generated by Toeplitz matrices, Functional Anal. Appl. 3 (1969), 119–137.Google Scholar
11. Power, S. C., C*-algebras generated by Hankel operators and Toeplitz operators, J. Functional Analysis, to appear.CrossRefGoogle Scholar
12. Sarason, D. E., On products of Toeplitz operators, ACTA Sci. Math. (Szeged) 35 (1973), 7–12.Google Scholar
13. Sarason, D. E., Toeplitz operators with semi-almost periodic symbols, Duke Math. j. 44 (1977), 357–364.Google Scholar
14. Sarason, D. E., Toeplitz operators with piecewise quasicontinuous symbol, Indiana Math. J. 26 (1977), 817–838.Google Scholar
15. Widom, H., Inversion of Toeplitz matrices II, Illinois J. Math. 4 (1960), 88–89.Google Scholar