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15.—The Measure of Non-compactness of Some Linear Integral Operators

  • C. A. Stuart (a1)

Synopsis

The measure of non-compactness of linear integral operators on the half-line [0, ∞) of a special type is studied. In particular, a necessary and sufficient condition is established for an operator of this type to define a compact operator from L2(0, ∞) into itself. These results are then used to discuss the spectrum of second-order differential operators. A necessary and sufficient condition for the spectrum to be discrete is established together with estimates for the distance of a point in the resolvent set from the essential spectrum.

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References

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Browder, F. E., 1961. On the spectral theory of elliptic differential operators, Math. Annln, 142, 22139.
Chisholm, R. S. and Everitt, W. N., 1971. On bounded integral operators in the space of integrable-square functions, Proc. Roy. Soc. Edinb., 69A, 199204.
Dunford, N. and Schwartz, J. T., 1963. Linear Operators, Part II. New York: Interscience.
Kato, T., 1966. Perturbation Theory for Linear Operators. Berlin: Springer.
Naimark, M. A., 1968. Linear Differential Operators, Part II. London: Harrap.
Nussbaum, R. D., 1970. The radius of the essential spectrum, Duke Math. J., 37, 473478.
Stuart, C. A., 1973. Some bifurcation theory for K-set contractions, Proc. Lond. Math. Soc. (in press).
Yosida, K., 1966. Functional Analysis (1st Edn.). Berlin: Springer.

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